How to Find Y Intercept with 2 Points

How to Find Y Intercept with 2 Points

How to Find Y Intercept with 2 Points

Step-by-Step Guide: How to Find y intercept with 2 points given to you

 

Free Step-by-Step Guide: How to find y-intercept of a graph given two points.

 

Understanding how to find the y-intercept of a line given 2 points (i.e. (x,y) coordinates) that the line passes through is an incredibly important and useful algebra skill that every student can easily learn with a little practice. In fact, knowing how to find the y intercept with 2 points given is a foundational skill that will help you to develop a stronger overall understanding of linear equations on the coordinate plane.

If you are an algebra student who is struggling with this key skill or an algebra teacher looking for a simple way to explain this topic in your classroom, this free step-by-step guide to finding y intercept with 2 points given shares everything you will ever need.

In the sections ahead, we will walk through all of the steps to finding the y intercept of a line with 2 points, including a recap of some key vocabulary as well as exactly how to solve any problem that gives you 2 coordinate points and asks you to find the y-intercept of a line. After working through different practice problems, you will have gained significant practice and experience with this key algebra skill.

While we recommend that you work through each section of this guide in order (the review section covers several important foundational skills and vocabulary terms related to linear functions), you can jump to any section by using the quick-links below. Ready to get started?

How to Find Y Intercept with 2 Points: Sections:

 

Figure 01: What is the Y-Intercept?

 

Quick Review: Lines and Y-Intercepts

Before we learn how to find y intercept with 2 points, let’s do a quick review of some key algebra terms and concepts related to linear functions.

For starters, it’s important to remember that linear equations can be expressed in slope-intercept form, also known as y = mx + b form, where:

  • slope-intercept form: y = mx + b

  • m is the slope

  • b is the y-intercept

For example, a line with the equation y = ⅓x + 4 has a slope of ⅓ and a y-intercept at 4.

What exactly is a y-intercept?

The y-intercept of a line is the coordinate point where the line crosses the y-axis. The y-intercept is always written as an (x,y) coordinate where x is 0.

So, the line y = ⅓x + 4 has a y-intercept at (0,4).

The image in Figure 01 above shows three different linear functions and their y-intercepts. Do you notice a pattern?

Knowing the coordinates of the y-intercept of a line is helpful because it tells you the starting point for graphing the line, and it also helps you to write the equation of the line in slope-intercept form, y=mx+b, where m is the slope of the line and b is the y-intercept.

 

Figure 02: Review: Slope-Intercept Form (y=mx+b)

 

So, whenever you are given two coordinate points that a line passes through, you have to figure out the values of m and b in order to determine the coordinates of the y-intercepts.

With these concepts in mind, let’s go ahead and try our first example problem of how to find y intercepts with 2 points.

Make sure that you know all of the perfect squares from 1 to 144 (as shown in Figure 01 above) before moving onto the next section.

While it is relatively easy to simplify a square root of a perfect square, it is more complicated to simplify a square root of a non-perfect square, which is what we will be focusing on in the next section, where, for example, we will look at finding the perfect squares of numbers like 18, 75, and 112.

How to Find Y Intercept with 2 Points

Now we are ready to try our first example where we will learn how to find y-intercept with 2 points given, and will solve them by following these 3 simple steps:

To find the y-intercept using two points, follow these steps:

Step 1: Use the Given Coordinates to Find the Slope of the Line

Step 2: Plug One point and Slope, m, into the Slope-Intercept Form Formula (y=mx+b)

Step 3: Solve for b and determine the y-intercept

Example A: Find the y intercept of the line passing through (2,5) and (4,9)

 

Example A: Find the y intercept of the line passing through (2,5) and (4,9).

For our first example, we are given two points and we are tasked with finding the y-intercept of the line that passes through them. We can use our 3-step strategy to solve this problem as follows:

Step 1: Use the Given Coordinates to Find the Slope of the Line

For this example, we are given the following coordinate points:

  • (2,5) and (4,9)

We can use the slope formula to calculate the slope of the line that passes through these points as follows:

  • m = (y₂ - y₁) / (x₂ - x₁)

  • m = (9-5) / (4-2) = 4/2 = 2

  • m=2

So, the line has a slope of 2.

Step 2: Plug One point and Slope, m, into the Slope-Intercept Form Formula (y=mx+b)

Now that we know that m=2, we can start to build the equation of this line in y=mx+b form as follows:

  • y = 2x + b

For the second step, we have to select one of the two (x,y) coordinates that was given to us, plug it into the slope-intercept form equation, and solve for b as follows:

  • Let’s choose the point (2,5), where x=2 and y=5

  • y = 2x + b

  • 5 = 2 (2) + b

  • 5 = 4 + b

  • 1 = b → b = 1

We have now solved for b, and we can conclude that b=1.

Step 3: Solve for b and determine the y-intercept

Now that we know that b=1, we can write the y intercept as a coordinate point and finish the problem.

Remember that the y-intercept of a line is always written as an (x,y) coordinate where x is 0.

Final Answer: The line has an equation of y=2x + 1 and the y-intercept is at (0,1)

Figure 02 shows the step-by-step process for solving this first problem, and Figure 03 shows the graph of y=2x+1 (notice how the line passes through the two given points (2,5) and (4,9) as well as the y-intercept at (0,1).

Figure 02: How to Find Y Intercept with 2 Points (Step-by-Step)

Figure 03: The graph of y=2x+1

Now that we have solved our first how to find y intercept with 2 points problem, let’s apply what we have learned to another example.

Example B: Find y intercept with 2 points given: (-1,4) and (3,0)

 

Example B: Find the y intercept of the line passing through (-1,4) and (3,0).

Let’s go ahead and solve this next problem using the same steps as the previous example.

Step 1: Use the Given Coordinates to Find the Slope of the Line

For Example B, we know that the line passes through the points:

  • (-1,4) and (3,0)

Let’s plug these points into the slope formula to figure out the slope (m) of the line as follows:

  • m = (y₂ - y₁) / (x₂ - x₁)

  • m = (0-4) / (3-(-1)) = 4/4 = -1

  • m=-1

This line has a slope of —1.

Step 2: Plug One point and Slope, m, into the Slope-Intercept Form Formula (y=mx+b)

Now we can write the equation of the line in slope-intercept form where m=-1 as follows:

  • y = -1x + b

  • y= -x + b

Now, let’s take one of the two given points and plug it into the slope-intercept form equation to solve for b as follows:

  • Let’s choose the point (3,0), where x=3 and y=0

  • y = -x + b

  • 0 = -(3) + b

  • 0 = -3 + b

  • 3 = b → b = 3

Therefore, we know that the y-intercept (b) is equal to 3.

Step 3: Solve for b and determine the y-intercept

Finally, we can take our result from Step 2 (b=3) and use it to write the y-intercept in coordinate form.

Final Answer: The line has an equation of y=-x + 3 and the y-intercept is at (0,3)

The entire process for solving this problem is shown in Figure 04 below, and the corresponding graph is shown in Figure 05.

Figure 04: How to find y-intercept with 2 points

Figure 05: The graph of y=-x+3.

 
 

More Free Math Resources:

Free Fraction to Decimal Chart (Printable PDF)

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Free Fraction to Decimal Chart (Printable PDF)

Image: Mashup Math

Free Fraction to Decimal Chart

Are you in need of a quick reference chart for converting between fractions and decimals? Whether you are a student learning how to convert between fractions and decimals, or someone who could use a handy reference for everyday use, the free fraction to decimal chart shared on this page is the perfect tool for you.

Having a fraction to decimal conversion chart is incredibly useful for making fast and one hundred percent accurate conversions between decimals and fractions. Our free fraction to decimal chart includes 64 common fractions and their decimal equivalents. In terms of measurement, our fractions to decimals chart includes conversions for all common fractional measurement units starting at 1/64.

In the next section, you will find a link to download your free Fraction to Decimal Chart as a PDF file. Some great ways to use our free fractions to decimals chart include:

  • Saving the PDF file on your phone to reference conversions wherever you go.

  • Print the chart and have it laminated for repeated use (you can keep it in your notebook, tool box, etc.)

  • Email the PDF chart to yourself and keep it in your inbox as an accessible conversion guide.

Do you want to learn how to convert between fractions and decimals without a chart? Check out our free step-by-step lesson on converting between decimals and fractions in 3 easy steps!

 

Free Fractions to Decimals Chart (Preview)

 

Fraction to Decimal Chart Download

Click Here to Download Your free Fraction to Decimal Conversion Chart PDF

When you click the link above, a PDF window will open where you can download and/or print our free decimals to fractions chart.

Decimal to Fraction Chart: Real-World Uses

Now that you have downloaded your free Fraction to Decimal Conversion Chart, here are few ideas for some awesome real-world applications of this super hand tool:

Cooking and Baking: Many recipes include fractional portions (e.g. ⅓ cup of water, ½ pound of sugar, etc.) as suggested portion sizes, while many kitchen tools (such as digital scales) use decimal measurements. To ensure that you are following any given recipe correctly and accurately, it is important to be familiar with fraction and decimal conversions. For example, knowing that ⅓ is roughly approximate to 0.333 will allow you follow recipes and use tools that incorporate fractions or decimals.

Money: Another useful application of our fraction to decimal chart is working with money. For example, if you know that the fraction ¼ is equal to the decimal 0.25, then you can easily determine and understand things like sales prices, discounts, taxes, and overall budgeting. If you want to improve your ability to work with money and make sound financial decisions, then being able to convert between fractions and decimals is a great foundational skill that will serve you for a lifetime.

Sports: Numerical statistics are a huge part of sports, and being able to accurately convert between fractions and decimals allows you to better make sense of sports-related data. For example, if a basketball team has won 6 out of 8 games (6/8 simplifies to 3/4), we can conclude that the team has a 75% winning percentage (because 3/4 equals 0.75).

 

Having a fractions to decimals chart is a handy tool for construction professionals. (Photo by Josh Olalde on Unsplash)

 

Fractions to Decimals Chart for Construction and Home Improvement Projects

Construction and Home Improvement: Any time that you have to measure the length of something, it is super important for you to be able to easily convert between fractions and decimals. In fact, most rulers are divided by segments that each represent 1/64 of an inch, and construction professionals have to be familiar with dozens of fraction to decimal conversions (which is why many of them purposely keep a fraction to decimal chart in their toolbox for quick reference.

For example, builders must know that ⅝ of an inch is the same as 0.625 inches in order to make correct measurements, create accurate scale drawings, and to ensure that their plans are build according to the design. This is why, in construction, there is a common saying: measure twice and cut once. This is because it is imperative to get your measurements correct before you make a cut (e.g. a peice of wood or metal). Why? Because, once you cut something to size, there are no do-overs. So, you absolutely have to get your measurements right before making cuts.

So, having a fractions to decimals chart on hand is a great tool that will serve you in a variety of situations inside and outside of the classroom. By having a quick and reliable reference guide at your disposal, you can spend more focus and energy on whatever project you are working on (baking, buying, building, etc.), and less time on making mathematical calculations and conversions between fractions and decimals.

Read More Posts About Math Education:

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How to Simplify a Square Root in 3 Easy Steps

How to Simplify a Square Root in 3 Easy Steps

How to Simplify a Square Root

Step-by-Step Guide: How to Simplify a Square Root in 3 Steps

 

Free Step-by-Step Guide: Ready to learn how to simplify a square root?

 

Simplifying square roots is a useful and important math skill that every student can learn with enough practice. By learning a simple 3-step process for simplifying square roots, you can learn to quickly and correctly simplify any square root (whether it is a perfect square or not), and that is exactly what we will be doing in this free guide.

The sections below will teach you exactly how to simplify a square root using a simple step-by-step method. Together, we will recap some key concepts and vocabulary terms and then work through three examples of how to simplify a square root. Whether you are learning this skill for the very first time or you are an experienced student in need of a quick and comprehensive review, this page will share everything you need to know about how to simplify a square root.

This guide is organized based on the following sections:

You can use the text links above to jump to any section of this guide, or you can work through the sections in order. Let’s get started!

 

Preview: How to Simplify a Square Root in 3 Steps.

 

Quick Intro: Square Roots and Perfect Squares

Before we work through examples of how to simplify a square root, let’s quickly recap some important concepts and vocabulary terms related to this topic.

In math, square roots are the inverse (or opposite) operation of squaring a number (i.e. multiplying a number by itself). And, conversely, the square root of a number is the value that, when multiplied by itself, results in the number that you started with.

For example, consider the square root of 16, which can be expressed using square root notation:

  • √16

We can say that √16 equals 4 because 4 times itself (i.e. 4x4 or 4²) equals 16, therefore:

  • √16 = 4 → because 4² = 16

Numbers like 16 are called perfect squares because their square roots are whole numbers, which makes them very easy to simplify.

In fact, you are likely already familiar with how to simplify many perfect squares such as:

  • √4 = 2 → because 2² = 4

  • √9 = 3 → because 3² = 9

  • √16 = 4 → because 4² = 16

  • √25 = 5 → because 5² = 25

  • √36 = 6 → because 6² = 36

  • √49 = 7 → because 7² = 49

  • √64 = 8 → because 8² = 64

  • √81 = 9 → because 9² = 81

  • √100 = 10 → because 10² = 100

Figure 01: Perfect Squares up to 144

Make sure that you know all of the perfect squares from 1 to 144 (as shown in Figure 01 above) before moving onto the next section.

While it is relatively easy to simplify a square root of a perfect square, it is more complicated to simplify a square root of a non-perfect square, which is what we will be focusing on in the next section, where, for example, we will look at finding the perfect squares of numbers like 18, 75, and 112.

How to Simplify a Square Root Examples

Now we are ready to use the following simple 3-step method for simplifying square roots to solve three practice problems:

Steps: How to Simplify a Square Root

  • Step 1: Identify two factors where one of them is a perfect square (choose the largest perfect square factor), and rewrite as a product.

  • Step 2: Split the product using two square root symbols.

  • Step 3: Simplify the perfect square and rewrite your final answer.

Example A: Simplify √18

 

Example A: Simplify √18

In our first example, we want to simplify a square root of a non-perfect square: √18.

We can simplify a square root like √18 by using our three step strategy as follows:

Step 1: Identify two factors where one of them is a perfect square.

Let’s start by listing the factors of 18:

  • Factors of 18: 1, 2, 3, 6, 9 and 18

Notice that 18 has one factor that is a perfect square: 9, and that:

  • 9 x 2 = 18

Since 9 x 2 equals 18, the two numbers that we are going to use for Step 2 are 9 and 2.

Step 2: Split the product using two square root symbols.

For the second step, we can use the factors from Step 1 to rewrite √18 as follows:

  • √18 = √(9 x 2) = √9 x √2

We can “split” the square root in this way because of the product property of square roots, which says that:

  • √(A x B) = √(A) x √(B)

(provided that A and B are non-negative numbers).

So we now have a new equivalent product that represents √18, which is…

  • √9 x √2

Step 3: Simplify the perfect square and rewrite your final answer.

For our final step, notice that one of the square roots in our new expression, √9, is a perfect square. Since we know that √9 = 3, we can rewrite √9 as 3 as follows:

  • √9 x √2 = 3 x √2

Now, all that we have to do is rewrite the result, 3 x √2, as 3√2 , and we have solved the problem!

Final Answer: √18 = 3√2

Figure 02 below shows the step-by-step process for simplifying this square root.

 

Figure 02: How to Simplify a Square Root in 3 Steps.

 

Now that you have learned how to simplify a square root, let’s gain some more experience by working through another example.

Example B: Simplify √75

 

Example B: Simplify √75

We can solve this next example by using the three steps that we used to solve the previous example.

Step 1: Identify the factors of 75 and determine the largest perfect square factor.

We can begin by listing the factors of 75:

  • Factors of 75: 1, 3, 5, 15, 25 and 75

Notice that 75 has one perfect square factor, 25, and that:

  • 25 x 3 = 75

Step 2: Split the product using two square root symbols.

Next, we can use the factors from Step 1 to rewrite √75as follows:

  • √75 = √(25 x 3) = √25 x √3

Step 3: Simplify and solve.

Finally, we can simplify √25 as 5 (since √25=5) and rewrite the expression as follows:

  • √25 x √3 = 5 x √3

We can now rewrite 5 x √3 as 5√3 and we can conclude that:

Final Answer: √75 = 5√3

The entire process for solving this problem is shown in Figure 03 below.

 

Figure 03: Multiplying fractions with whole numbers Example B solved.

 

Example C: Simplify √112

 

Example C: Simplify √112

For our final step-by-step example of how to simplify a square root, let’s take on a triple-digit number using our three-step method.

Step 1: Identify the factors of 112 and pick out the largest perfect square factor.

  • Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56 and 112

Notice 112 has two perfect square factors: 4 and 16. In cases like this, always choose the largest perfect square factor (16 in this case).

Now that we have identified our perfect square factor, we can say that:

  • w16 x 7 = 112

Step 2: Split the product using two square root symbols.

Next, we can write √112 as follows:

  • √112 = √(16 x 7) = √16 x √7

Step 3: Simplify and solve.

For the final step, we can simplify √16 as 4 and rewrite the expression as follows:

  • √16 x √7 = 4 x √7

Now we just have to rewrite 4 x √7 as 4√7 and we have solved the problem!

Final Answer: √112 = 4√7

The three-step process for solving Example C is shown in Figure 03 below

 

Figure 03: How to Simplify a Square Root in 3 Steps.

 
 
 

More Free Math Resources:

How to Multiply Fractions with Whole Numbers—Step-by-Step

How to Multiply Fractions with Whole Numbers—Step-by-Step

How to Multiply Fractions with Whole Numbers

Step-by-Step Guide: How to Multiply Fractions with Whole Numbers, Multiplying Fractions by Whole Numbers Examples

 

Free Step-by-Step Guide: How to multiply fractions with whole numbers explained.

 

Multiplying fractions with whole numbers can seem like a challenging math skill, but, with some simple strategies and an easy step-by-step method, it can be a relatively easy task that any student can master.

In this free guide, we will work through several examples of how to multiply fractions with whole numbers using a simple step-by-step process. As long as you can follow three easy steps, you will be able to confidently and accurately solve a variety of math problems where you have to multiply fractions with whole numbers.

You can work through the sections in this free guide in sequential order, or you can click on any of the quick-links below to jump to one particular section.

Quick Intro: Multiply Fractions with Whole Numbers

Before we dive into any examples of how to multiply fractions with whole numbers, let’s do a quick introductory review of what it means when we multiply fractions with whole numbers.

For example, let’s consider the example 3 x 1/4:

  • 3 is the whole number

  • 1/4 is the fraction

Whenever you multiply a fraction by a whole number, you are really just performing repeated addition (i.e. you are adding the fraction to itself a number of repeated times that is determined by the whole number).

If we think of multiplication in terms of repeated addition, we can rewrite 3 x 1/4 as follows:

  • 3 x 1/4 = 1/4 + 1/4 + 1/4

And, since 1/4 + 1/4 + 1/4 is equal to 3/4, we can conclude that:

  • 3 x 1/4 = 1/4 + 1/4 + 1/4 = 3/4

Final Answer: 3 x 1/4 = 3/4

 

Figure 01: How to Multiply Fractions with Whole Numbers Using Repeated Addition.

 

The process of multiplying fractions with whole numbers using repeated addition is shown in Figure 01.

While we will not use repeated addition to solve the examples in this guide, understanding this basic relationship between multiplication and repeated addition is the first step to easily learning how to multiply fractions with whole numbers.

Now, let’s go ahead and work through some examples of multiplying fractions with whole numbers using a simple 3-step method.

Multiplying Fractions by Whole Numbers Examples

For all of the multiplying fractions with whole numbers examples that follow, we will be using the following 3-step method for solving:

  • Step 1: Rewrite the whole number as a fraction with a denominator of 1.

  • Step 2: Multiply the numerators together and then multiply the denominators together.

  • Step 3: Simplify if possible.

 

Example A: Multiplying whole numbers with fractions.

 

Example A: Multiply 2 x 1/3

For our first example, we have to multiply the whole number 2 by the fraction ⅓, and we will do that by following our 3-step process as follows:

Step 1: Rewrite the whole number as a fraction with a denominator of 1.

First, we can rewrite the whole number, 2, as a fraction with a numerator of 1 as follows:

  • 2 → 2/1

Now we have a new multiplication problem:

  • 2 x 1/3 → 2/1 x 1/3

Step 2: Multiply the numerators together and then multiply the denominators together.

Now that we have a new expression with two fractions being multiplied by each other, we can perform multiplication by multiplying the numerators together and then multiplying the denominators together as follows:

  • 2/1 x 1/3 = (2x1) / (1x3) = 2/3

After completing Step 2, we are left with the fraction 2/3.

Step 3: Simplify if possible.

Finally, we just have to check if our result from Step 2, 2/3, can be simplified.

In this case, the fraction 2/3 can not be simplified because there is no common factor between the numerator (2) and the denominator (3) other than 1.

Final Answer: 2 x 1/3 = 2/3
The complete step-by-step process for solving this first example is shown in Figure 02 below.

 

Figure 02: How to multiply fractions with whole numbers step-by-step.

 

Now that you are familiar with our 3-step method for multiplying fractions with whole numbers, let’s gain some more experience by using them to solve another example.

 

Example B: 2/3 × 34 = ?

 

Example B: Multiply 2/3 x 4

For this next example, notice how, in this case, the first term is a fraction and the second term is the whole number (this is a reverse situation compared to Example A). However, the commutative property of multiplication tells us that the order of the terms does not matter, so we can still use our 3-step process to solve this problem as follows:

Step 1: Rewrite the whole number as a fraction with a denominator of 1.

We can leave the fraction 2/3 alone and rewrite the whole number 4 as a fraction with a denominator of 1 as follows:

  • 2/3 x 4 → 2/3 x 4/1

Step 2: Multiply the numerators together and then multiply the denominators together.

Next, we can take our new expression and simply multiply the numerators together, and then the denominators together as follows:

  • 2/3 x 4/1 = (2x4) / (3x1)

  • (2x4) / (3x1) = 8/3

After completing the second step, our result is 8/3. We can now move onto the third and final step.

Step 3: Simplify if possible.

Let’s see if our result from Step 2, 8/3, can be simplified.

Since there is no common factor (other than 1) between the numerator (8) and the denominator (3), we know that the fraction 8/3 can not be simplified. However, since 8/3 is an improper fraction, we do have the option of either expressing it as 8/3 or as the mixed number 2 2/3 (in this case, we will choose to express our answer as 8/3).

Final Answer: 2/3 x 4 = 8/3
Figure 03 below illustrates our step-by-step process for solving this second example.

 

Figure 03: Multiplying fractions with whole numbers Example B solved.

 
 

5Example C: 10 × 1/5 = ?

 

Example C: Multiply 10 x 1/5

Let’s gain some more practice using our 3-step method for multiplying fractions with whole numbers.

Step 1: Rewrite the whole number as a fraction with a denominator of 1.

In this example, we have to rewrite the whole number (10) as a fraction with a denominator of 1.

  • 10 x 1/5 → 10/1 x 1/5

Step 2: Multiply the numerators together and then multiply the denominators together.

Next, let’s take our new expression, 10/1 x 1/5, and multiply the numerators and denominators together:

  • 10/1 x 1/5 = (10x1) / (1x5)

  • (10x1) / (1x5) = 10/5

Finally, let’s move onto Step 3 to see if our result (10/5) can be simplified.

Step 3: Simplify if possible.

In this case, the numerator (10) and the denominator (5) share a common factor of 5. So, we can simplify 10/5 by dividing both the numerator and denominator by 5 as follows:

  • 10 ÷ 5 = 2

  • 5 ÷ 5 = 1

After dividing, we can say that 10/5 = 2/1, and we can rewrite 2/1 as 2.

Final Answer: 10 x 1/5 = 10/5 = 2/1 = 2
The entire process of solving Example C is shown in Figure 04 below illustrates our step-by-step process for solving this second example.

 

Figure 05: Example C Solved.

 
 

Example D: 7 × 5/6 = ?

 

Example D: Multiply 7 x 5/6

Let’s work through one final example of multiplying fractions with whole numbers.

Step 1: Rewrite the whole number as a fraction with a denominator of 1.

In this case, we can rewrite the whole number (7) as a fraction as follows:

  • 7 x 5/6 → 7/1 x 5/6

Step 2: Multiply the numerators together and then multiply the denominators together.

Next, we can multiply the two fractions together as follows:

  • 7/1 x 5/6 = (7x5) / (1x6)

  • (7x5) / (1x6) = 35/6

Step 3: Simplify if possible.

Finally, we have to see if our result from Step 2 (35/6) can be simplified. Since 35 and 6 do not share any common factors besides 1, we know that it can not be simplified any further.

Final Answer: 7 x 5/6 = 35/6
Figure 06 shows how we solved this final example.

 

Figure 06: How to Multiply Fractions with Whole Numbers.

 
 
 

More Free Math Resources:

How to Factor Polynomials—Step-by-Step Examples and Tutorial

How to Factor Polynomials—Step-by-Step Examples and Tutorial

How to Factor Polynomials Explained

Factoring polynomials is a process of rewriting a polynomial as the product of one or more simpler expressions—including constants, variables, or factors that can not be further reduced. How to factor a given polynomial will depend on a few different factors, including the number of terms, the value of the coefficients, and the structure of the polynomial.

 

Free Step-by-Step Guide: How to factor a polynomial with a specific number of terms

 
Anthony Persico

by Anthony Persico

Founder & Head Educator, Mashup Math
With a background in elementary and secondary math education and special education, Anthony has taught thousands of K-12 students across the United States.

Learn more about the author →

Last Updated: May 2025

In algebra, a polynomial is an expression made up of variables and coefficients separated by the operations of addition and/or subtraction.

Polynomials are a fundamental math topic and understanding how to work with them (including factoring) is essential to being successful in algebra and beyond. Learning how to factor polynomials with 2, 3, or 4 terms involves understanding how to break down a given polynomial into simpler factors.

This free Step-by-Step Guide on How to Factor Polynomials will cover the following topics:

Table of Contents

While learning how to factor polynomials can be challenging, it is a learnable skill that can be acquired through practice. The goal of this free guide on how to factor polynomials is to give you plenty of step-by-step practice with factoring polynomials—including polynomials with 4 terms (cubic polynomials)—so that can become more comfortable with factoring all kinds of polynomials.

Before we cover everything you need to know about how to factor a polynomial, let’s quickly recap some key algebra vocabulary terms and phrases that you will need to be familiar with in order to use this guide.

 

Figure 01: How to Factor Polynomials: What is a Polynomial?

 

What is a polynomial?

As previously stated, a polynomial is a math expression comprised of variables, coefficients, and/or constants separated by the operations of addition or subtraction.

The terms of polynomials are individual parts, or monomials, separated by addition or subtraction signs.

For example,

  • 3x² is a monomial

  • 3x² + 6x is a polynomial with 2 terms (3x² and 6x)

  • 3x² + 6x - 15 is a polynomial with 3 terms (3x², 6x, and -15)

  • 9x³ + 3x² + 6x - 15 is a polynomial with 4 terms (9x³, 3x², 6x, and -15)

Figure 01 above illustrates the difference between a monomial and a polynomial.

For an expression to be considered a polynomial, it must have at least two terms, but there is no limit on how many terms a polynomial can have.

When it comes to factoring polynomials, you will most commonly be dealing with polynomials that have 2 terms, 3 terms, or 4 terms:

  • A polynomial with 2 terms is called a binomial

  • A polynomial with 3 terms is called a trinomial

  • A polynomial with 4 terms is called a quadrinomial (also known as a cubic polynomial)

Examples of a polynomial with 2 terms, 3 terms, and 4 terms are shown in Figure 02 below.

 

Figure 02: How to factor polynomials with 4 terms or less.

 

Now that you understand the key terms and the difference between a polynomial with 2 terms, 3 terms, and 4 terms.

For factoring each type of polynomial, we will look at two methods: GCF, direct factoring, and a combination of the two.

Let’s get started!

How to Factor Polynomials with 2 Terms

We will start by learning how to factor polynomials with 2 terms (binomials).

Whenever you are factoring a polynomial with any number of terms, it is always best to start by looking to see if there is a GCF—or greatest common factor—that all of the terms have in common.

For example, consider the following example:

Example #1: Factor 8x + 4

For this example, you should notice that both terms, 8x and 4 are divisible by 4, hence they share a GCF of 4.

Therefore, you can divide out the GCF of 4 from both terms as follows:

  • 8x + 4 → 4 (2x + 1)

So, the factors of 8x + 4 are: 4 and (2x+1).

What we just did was essentially the reverse of the distributive property, as shown in Figure 03 below.

 

Figure 03: How to factor a polynomial with 2 terms using the GCF method.

 

Note that many binomials can be factored using the GCF method, so let’s gain a little more practice with one more example (understanding how to simplify and/or factor a polynomial using the GCF method will come in handy when you start factoring 3 and 4 term polynomials later on).

Example #2: Factor 6x² + 12x

Just like the first example, there is a GCF for both terms. But, in this case, the GCF includes a variable. Why? Because both terms have coefficients that are divisible by 6 and both terms have at least one x variable, so the GCF, in this case, is 6x.

Therefore, you can divide out 6x from both terms as follows:

  • 6x² + 12x → 6x(x + 2)

So, the factors of 6x² + 12x are: 6x and (x+2).

Again, this method of factoring is just the reverse of the distributive property and is illustrated in Figure 04 below.

 

Figure 04: How to factor a polynomial with 2 terms using the GCF method.

 

Next, we will look at a special case of factoring a binomial—when the binomial is a difference of two squares (this is sometimes referred to as DOTS).

Whenever you have a binomial of the form a²-b², the factors will be of the form (a+b)(a-b).

Example #3: Factor x² - 49

For example, if you wanted to factor the binomial: x² - 49, you would notice that both x² and 49 are squares:

  • x² = (x)(x)

  • 49=(7)(7)

So, another way to write (x²- 49) is (x²- 7²)

Therefore, you can use the DOTS method for factoring binomials. In this case, a = x and b = 7, so:

  • (a²-b²) = (a+b)(a-b) → (x²- 7²) = (x-7)(x+7)

You can now conclude that the factors of x²- 49 are (x-7) and (x+7) using the DOTS method.

This process is illustrated in Figure 05 below:

 

Figure 05: How to factor a polynomial that is the difference of two squares.

 

If you want to learn more about the DOTS method for factoring polynomials that are the difference of two squares, check out this free video tutorial on YouTube for more practice.

And if you want more practice, check out our free Factoring Polynomials Worksheet Library, where you can download free PDF practice worksheets with answer keys.

Otherwise, let’s continue onto the next section where you will learn how to factor polynomials with 3 terms.

How to Factor Polynomials with 3 Terms

Moving on, we will now look at polynomials with 3 terms, typically referred to as trinomials

Learning how to factor polynomials with 3 terms involves a more involved factoring process that we will explore in this section.

The trinomials that we will cover will be of the form ax² + bx + c (where c is a constant). The strategies that we will use will depend on whether a (the leading coefficient) equals one or not. Therefore, the first two examples in this section will be factoring trinomials when a=1 and the second two examples will be when a≠1.

How to Factor Polynomials with 3 Terms when a=1

Example #1: Factor x² + 6x + 8

For the first example, we have to factor the trinomial: x² + 6x + 8

 

Figure 06: How to factor polynomials with 3 terms (when a=1)

 

Again, the leading coefficient, a, is equal to 1 in this example. This is important to note because the following method for factoring a trinomial only works when a=1.

Now we are ready to factor this trinomial in 3 easy steps:

Step One: Identify the values of b and c.

In this example, the values of b and c in the trinomial are: b=6 and c=8

Step Two: Figure out two numbers that both ADD to b and MULTIPLY to c.

The second step often involves some of trial-and-error as you pick numbers and see if they meet both conditions (the two numbers have to add together to make b and multiply together to make c).

  • 5 + 1 =6 (the value of b) ✓

  • 5 x 1 ≠ 8 (the value of c) ✘

For example, lets say that you chose the numbers 5 and 1. While 5+1=6 is true (satisfying the first condition), 5x1=5 (not 8), therefore, they do not satisfy the second condition. So, 5 and 1 do not work.

But, if you picked the numbers 2 and 4, you can see that:

  • 2 + 4 =6 (the value of b) ✓

  • 2 x 4 = 8 (the value of c) ✓

Since 2 and 4 satisfy both conditions, you can stop searching and move onto the third step.

Step Three: Use your numbers from step two to write out the factors

In this case, you can conclude that the factors of x² + 6x + 8 are (x+2) and (x+4).

 

Figure 07: The factors of x² + 6x + 8 are (x+2) and (x+4).

 

You can verify that these are the correct factors by performing double distribution as follows:

  • (x+2)(x+4) = x² + 2x + 4x + 8 = x² + 6x + 8

Notice that you ended up with the trinomial that you started with! Now, lets work through one more example of how to factor polynomials with 3 terms when a=1.

Example #2: Factor x² - 3x - 40

For this next example, we have to factor the trinomial: x² - 3x - 40

 

Figure 08: How to factor 3rd degree polynomials

 

Notice that, in this case, the trinomial includes subtraction signs, which will affect how you perform step two below.

Step One: Identify the values of b and c.

For this trinomial, b= -3 and c= -40

Step Two: Figure out two numbers that both ADD to b and MULTIPLY to c.

Again, you have to find two numbers that add to make -3 and that multiply together to make -40.

This part can be tricky when both of the values for b and c are negative (like in this example). You have to recall that a negative number times another negative number will lead to a positive result, so you can’t have two negatives (since you need to find two numbers that multiply together to make -40).

Eventually, after some trial-and-error, you should find that -8 and +5 satisfy both conditions:

  • -8 + 5 =-3 (the value of b) ✓

  • -8 x 5 = -40 (the value of c) ✓

Step Three: Use your numbers from step two to write out the factors

Finally, you can conclude that the factors of x² - 3x -40 are (x-8) and (x+5).

(You make sure that this answer is correct, you can perform double distribution on (x-8)(x+5) to make sure that the result is equal to the original trinomial).

 

Figure 09: The factors of x² - 3x -40 are (x-8) and (x+5).

 

If you want more practice factoring trinomials when a=1, check out our free step-by-step guide on how to factor trinomials to gain some more practice.

And if you want more independent practice opportunities, check out our free Factoring Polynomials Worksheet Library, where you can download free PDF practice worksheets with answer keys.

Otherwise, you can continue on to learn how to factor polynomials with 3 terms when a≠1.

How to Factor Polynomials with 3 Terms when a≠1

Example #1: Factor 2x² - x - 6

For the first example, we have to factor the trinomial: 2x² - x - 6

 

Figure 10: How to factor polynomials with 3 terms when a≠1

 

For starters, notice that you can not pull out a GCF.

So, to solve trinomials of the form ax² + bx + c when a≠1, you can use the AC method as follows:

Step One: Identify the values of a and c and multiply them together

In this case, a=2 and c=-6, so

  • a x c = 2 x -6 = -12

Step Two: Factor and replace the middle term

The second step requires you to use the result from step one to factor and replace the middle term.

The middle term is currently -1x and note that:

  • -12 = -4 x 3; and

  • -4 + 3 = -1

We chose -4 and 3 as factors because the sum of -4 and 3 equals negative 1, so we can rewrite the original trinomial as 2x² - 4x +3x - 6

 

Figure 11: Factor and replace the middle term

 

Step Three: Split the new polynomial down the middle and take the GCF of each side

Note that we are now working with a polynomial that actually has four terms: 2x² - 4x + 3x - 6

In this third step, you have to split the polynomial down the middle to essentially create two separate binomials that you can simplify by dividing GCF’s out of as follows:

  • First Half: 2x² - 4x = 2x(x-2)

  • Second Half: 3x - 6 = 3(x-2)

This third step is illustrated in Figure 12 below:

 

Figure 12: Split the new polynomial down the middle and take the GCF of each side

 

Step Four: Identify the Factors

Finally, you are ready to identify the factors.

The result from the previous step was 2x(x - 2) + 3(x -2). Hidden within this expression are your two factors, which you can see by looking at Figure 13 below.

 

Figure 13: The final step is to identify the factors

 

Finally, you can conclude that the factors of 2x² - x - 6 are (2x+3) and (x-2).

Clearly, factoring a trinomial when a≠1 can be a tricky and there are several steps along the way, but, the more that you practice this process, the better you will become at factoring polynomials with 3 terms like the one in this past example. To give you a little more practice, lets work through one more example before we move on to learning how to factor cubic polynomials.

Example #2: Factor 4x² - 15x + 9

 

Figure 14: Factor the trinomial where a=4, b=-15, and c=9

 

Step One: Identify the values of a and c and multiply them together

In this example, a=4 and c=9, so

  • a x c = 4 x 9 = 36

Step Two: Factor and replace the middle term

For the next step, note that the middle term is -15x, so you will need to find two numbers that multiply to 36 and add to -15:

  • 36 = -12 x -3; and

  • -12 + -3 = -15

Now, we can rewrite the original trinomial as 4x² -12x -3x +9

Step Three: Split the new polynomial down the middle and take the GCF of each side

For step three, you have to split the polynomial into two separate binomials and divide a GCF out of each one as follows:

  • First Half: 4x² -12x = 4x(x-3)

  • Second Half: -3x+9 -3(x-3)

Step Four: Identify the Factors

The last step is to identify the factors as shown in Figure 15 below.

 

Figure 15: The factors are (4x-3)(x-3)

 

Now, you can conclude that the factors of 4x² - 15x + 9 are (4x-3) and (x-3).

You can again use double distribution on (4x-3)(x-3) to verify that your solution is correct.

If you need more step-by-step help with how to factor polynomials with 3 terms when a does not equal 1, visit out our free Factoring Polynomials Worksheet Library, where you can download free PDF practice worksheets with answer keys.

Otherwise, continue on to the final section where you will learn how to factor polynomials with 4 terms.

How to Factor Polynomials with 4 Terms

The last section of this guide will cover how to factor polynomials with 4 terms and how to factor cubic polynomials.

In this section, we are going to apply a grouping method for how to factor a cubic polynomial that is very similar to the way that you factored trinomials when the leading coefficient, a, did not equal one in the last section. So, you may want to review that section before moving onto the 4 term polynomial factoring examples, however, it is not completely necessary, as we will be taking a step-by-step approach to solving two examples of factoring cubic polynomials.

Now, lets go ahead and work through our first example on how to factor cubic polynomials.

 

Figure 16: Polynomials with 4 terms are referred to as cubic polynomials.

 

Example #1: Factor 2x³ - 3x² + 18x - 27

For the first example, we have to factor the cubic polynomial: 2x³ - 3x² + 18x - 27

Step One: Split the cubic polynomial into groups of two binomials.

To factor this 4 term polynomial, we are going to apply what is called the grouping method, which requires you to split the polynomial into two groups (two separate binomials) with the goal of factoring a GCF out of each one.

 

Figure 17: How to Factor Cubic Polynomials by Grouping: The first step is to split the polynomial into two groups of binomials.

 

Remember that the goal is to create two separate binomials that have a GCF. If there is no apparent GCF, you have the option of swapping the positions of the middle terms (- 3x² and 18x), but that is not necessary for factoring this 4 term polynomial.

In this example, by the end of step one, you now have two groups to factor:

  • (2x³ - 3x²)

  • (18x - 27)

Step Two: Factor each binomial by pulling out a GCF

Now, go ahead and divide a GCF out of each binomial as follows:

  • (2x³ - 3x²) → x²(2x - 3)

  • (18x - 27) → 9(2x - 3)

This step is illustrated in Figure 18 below.

 

Figure 18: How to factor a cubic polynomial by grouping.

 

Step Three: Identify the factors

Notice that both results have a (2x-3) term. This is important and expected. If both results do not share a same term, then you either made a mistake or the polynomial with 4 terms is not factorable.

But, since we were able to factor each group by pulling out a GCF that resulted in both groups sharing a common factor of (2x-3), we know that we can factor out the other terms (x² and +9), so now have our factors: (x²+9) and (2x-3)

Final Answer: The factors of 2x³ - 3x² + 18x - 27 are (x²+9) and (2x-3)

The entire process of how to factor polynomials by 4 terms by grouping is illustrated in Figure 19 below.

 

Figure 19: The factors of 2x³ - 3x² + 18x - 27 are (x²+9) and (2x-3)

 

Example #2: Factor 3y³ + 18y² + y + 6

Let’s gain some more practice with how to factor a cubic polynomial by grouping by solving one more example problem.

In this case, we have to factor the cubic polynomial 3y³ + 18y² + y + 6 using the same grouping method as the previous example.

Step One: Split the cubic polynomial into groups of two binomials.

Start by splitting the cubic polynomial into two groups (two separate binomials).

 

Figure 20: Split the cubic polynomial into two groups of binomials and check to see if they can both be factored by pulling out a GCF.

 

As shown in Figure 20 above, by completing step one, you are left with these two groups

  • (3y³ +18y²)

  • (y+6)

Hold on! Before moving onto the next step, you should notice that the second group (y+6) cannot be factored by pulling out a GCF (because there is no greatest common factor between 1y and 6).

However, notice that we can swap the middle terms of the cubic polynomial (18y² and +y) as shown in Figure 21 below.

Now, we can factor a new 4 term polynomial 3y³ + y + 18y² + 6 that is equivalent to the original 4 term polynomial since the commutative property of addition allows you to rearrange the terms.

 

Figure 21: The commutative property of addition allows you to rearrange the middle terms when you are unable to group and GCF the original cubic polynomial.

 

Notice that you can split this new polynomial into two binomials that can be factored by pulling out a GCF:

  • (3y³ + y)

  • (18y² + 6)

 

Figure 22: After rearranging the original cubic polynomial, you can split it into two binomial groups that can be factoring by pulling out a GCF.

 

Step Two: Factor each binomial by pulling out a GCF

As illustrated in Figure 22 above, after rearranging the original cubic polynomial, you can split it into two binomial groups that can be factoring by pulling out a GCF as follows:

  • (3y³ + y) → y(3y² + 1)

  • (18y² + 6) → 6(3y² + 1)

Step Three: Identify the factors

Now, you can see that both factors have a (3y² + 1) term, which means that you have factored correctly.

Final Answer: The factors of 3y³ + 18y² + y + 6 are (y+6) and (3y² + 1)

The entire process of how to factor polynomials a cubic polynomial like the one in this example is illustrated in Figure 23 below.

 

Figure 23: How to factor cubic polynomials by grouping (step-by-step).

 

Factoring Polynomials Advanced Problems

Now that we have covered how to factor polynomials in a variety of different ways, let’s take a look at two advanced factoring polynomials problems. These are the type of questions that you might find on a unit test or a standardized exam. Let’s see if we can apply what we learned earlier in this guide to solve it.

Advanced Factoring Polynomials Problem #1

Factor: (2/3)x² + (1/3)x -1

At first glance, this factoring problem looks different than our previous examples because it has fractions as coefficients. However, since the polynomial is in ax² +bx + c form, we can factor it using a familiar strategy.

First, let’s identify the values of a, b, and c:

  • a= 2/3

  • b=1/3

  • c=-1

Since 2/3, 1/3, and 1 do not share any common factors, we can not pull out a GCF.

Figure 24: Factoring Polynomials Advanced Problem #1

But, we can use the AC method to factor this trinomial as follows:

Step One: Identify the values of a and c and multiply them together

In this case, a=2/3 and c= -1, so

  • a x c = 2/3 x -1 = (-2/3)x

Step Two: Factor and replace the middle term

For the next step, we have to use our result from step one to factor and replace the middle term (1/3)x:

The middle term is currently -1x and note that:

  • (1/3)x = (-2/3)x + 1x

Since our current middle term ((1/3)x is equivalent to (-2/3)x + 1x, we can rewrite the original trinomial as:

  • (2/3)x² - (2/3)x + 1x -1

Step Three: Split the new polynomial down the middle and take the GCF of each side

For the third step in the AC method, we have to split the polynomial down the middle and take the GCF of each side:

  • First Half: (2/3)x² - (2/3)x

  • Second Half: 1x -1

After taking out the GCF, we are left with:

  • First Half: (2/3)x² - (2/3)x → (2/3)x(x-1)

  • Second Half: 1x -1 → +1(x-1)

Step Four: Identify the Factors

Finally, you just have to identify the factors, which are ((2/3)x+1)(x-1)

Final Answer: (2/3)x² + (1/3)x -1 = ((2/3)x+1)(x-1)

The entire step-by-step process for solving this problem is shown in Figure 24.

 

Advanced Factoring Polynomials Problem #2

Factor: x³ -3x² -4x + 12

We can factor this cubic polynomial by grouping:

  • (x³ -3x²) + (-4x + 12)

Next, we have to factor each separate group:

  • (x³ -3x²) → x²(x-3)

  • (-4x + 12) → -4(x-3)

Now, we can factor out (x-3), and rewrite the result as:

  • (x²-4)(x-3)

While this may look like a final answer, we should notice that (x²-4) is a difference of two squares that we can factor as:

  • (x²-4) = (x+2)(x-2)

Finally, we can conclude that:

Final Answer: x³ -3x² -4x + 12 = (x-3)(x+2)(x-2)

Free Factoring Polynomials Practice Worksheets

If you want to further test your understanding of factoring polynomials, including the difference of two squares, factoring trinomials, factoring by completing the square, and factoring polynomials by grouping, then visit the Mashup Math Factoring Polynomials Worksheet Library, where you can download several free PDF practice worksheets with complete answer keys.

Click here to access our free Factoring Polynomials Worksheet Library.

How to Factor Polynomials: Conclusion

Learning how to factor a polynomial is an important algebra skill that every math student must learn at some point.

While factoring polynomials can be tricky, there are several useful and effective strategies that you can use to factor polynomials. The strategy that you choose will depend on how many terms a polynomial has (as you will often be dealing with factoring polynomials with 2, 3, or 4 terms).

The best way to get better at factoring polynomials (especially cubic polynomials that have 4 terms) is by working through practice problems step-by-step. If you feel like you need more practice, we highly recommend working through the examples in this guide several times to gain more experience.

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