Subjects

Algebra 2

Factoring

Greatest Common Factor

296

•

Dec 5, 2025

•

5 pages

Key Methods for Factorizing Algebraic Expressions

Layla Guthrie

@laylaguthrie_aoly

Factorization is a key algebra skill that helps simplify complex... Show more

1 / 5

# METHODS OF
# FACTOR IZATION
Idenifying the GCF
Find the GCF
a. 24 and 36
```
2/24 2/36
2/12 2118
2/6 319
2
3
```
Factors of 24 2.2 23
Fact

Methods of Factorization: Finding the GCF

Finding the Greatest Common Factor (GCF) is your first step in factoring expressions. It's like identifying what several terms share in common.

To find the GCF of numbers, break them down into their prime factors. For example, with 24 and 36, write 24 = 2×2×2×3 and 36 = 2×2×3×3. The common factors (2×2×3 = 12) give us the GCF.

For algebraic expressions like 7x³, 14x², and 21x⁴, find the GCF in two parts:

Find the GCF of the coefficients (7 in this case)
Find the variable with the lowest power (x² here) So the GCF is 7x².

Pro Tip: Always look for the GCF first before trying other factoring methods - it makes the next steps much easier!

When expressions contain binomials like 3x $a+b$ and 2y $a+b$ , the common factor is simply $a+b$ . Remember that your goal is to pull out everything that's common to all terms.

Factoring Out the GCF in Complex Expressions

Factoring gets more interesting with longer expressions. The key is to identify what's common to all terms and pull it out.

With expressions like 15y³+12y⁴, start by finding the GCF - here it's 3y³. Then rewrite as 3y³ $5+4y$ . For 9a⁴b-18a⁵b+27ab, the GCF is 9ab, giving us 9ab $1-2a+3a²$ .

When you see expressions like 2w $x+3$ -5 $x+3$ , look for the common binomial factor. Here $x+3$ appears in both terms, so we can factor it out: $x+3$ $2w-5$ .

More complex examples require careful grouping. For 8ax+12a+2bx+8b, you might not immediately see a common factor. Try regrouping: 4a $2x+3$ +2b $x+4$ , which reveals $x+4$ is common to both groups, giving the final factored form $3a+2b$ $x+4$ .

Remember: Sometimes you need to factor multiple times. In 16w⁴-40w³-12w²+30w, first factor out 2w to get 2w $8w³-20w²-6w+15$ , then look for other patterns in what remains.

Factoring Trinomials and Special Products

The A.C. method is super helpful for factoring trinomials like ax²+bx+c:

Find the product of a and c (that's ac)
Look for two numbers that multiply to give ac and add up to b
Split the middle term using these numbers
Group and factor

For example, with 2x²+7x+6:

ac = 12
We need numbers that multiply to 12 and add to 7 (that's 3 and 4)
Rewrite as 2x²+4x+3x+6
Factor by grouping: $2x+3$ $x+2$

Difference of squares follows the pattern a²-b²= $a+b$ $a-b$ . So y²-25 = $y+5$ $y-5$ . But remember: sum of squares $like p²+9$ cannot be factored with real numbers.

Watch out! Don't try to factor sum of squares like a²+b² - they're prime over real numbers and can't be factored further!

For complex expressions like w⁴-81, recognize that it's (w²)²-9² and factor it as $w²+9$ $w²-9$ , then factor w²-9 again to get $w²+9$ $w+3$ $w-3$ .

Perfect Square Trinomials and Advanced Factoring

Perfect square trinomials follow these patterns:

a²+2ab+b² = $a+b$ ²
a²-2ab+b² = $a-b$ ²

To spot them, check if:

The first and last terms are perfect squares
The middle term equals 2ab or -2ab

For example, x²+14x+49 is a perfect square trinomial because:

First term is x² (perfect square)
Last term is 49 = 7² (perfect square)
Middle term is 14x, which equals 2(x)(7)
Therefore, x²+14x+49 = $x+7$ ²

Another example: 25y²-20y+4 = $5y-2$ ² because:

25y² = (5y)²
4 = 2²
-20y = -2(5y)(2)

When factoring complex expressions like 18c³-48c²d+32cd², look for a common factor first. Here we have 2c, so we get 2c $9c²-24cd+16d²$ , which reveals a perfect square trinomial: 2c $3c-4d$ ².

Smart approach: Always check if your trinomial might be a perfect square - it makes factoring much faster than using other methods!

Sum and Difference of Cubes

The sum of cubes follows the pattern: a³+b³ = $a+b$ $a²-ab+b²$

The difference of cubes follows: a³-b³ = $a-b$ $a²+ab+b²$

These patterns are super useful for expressions like w³+64:

Recognize that 64 = 4³
Apply the formula: w³+4³ = $w+4$ $w²-4w+16$

For more complex examples like 27p³-1000q³:

Rewrite as (3p)³-(10q)³
Apply the difference of cubes formula: $3p-10q$ $9p²+30pq+100q²$

These factoring techniques work together. For 3y⁴-48:

Factor out common factor: 3 $y⁴-16$
Recognize y⁴-16 as a difference of squares: $y²-4$ $y²+4$
Factor y²-4 as another difference of squares: $y-2$ $y+2$
Final answer: 3 $y-2$ $y+2$ $y²+4$

Factoring strategy: When you see expressions with cubes or higher powers, think about using these special patterns - they save a ton of work compared to other methods!

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Subjects

Algebra 2

Factoring

Greatest Common Factor

Algebra 2

•

296

•

Dec 5, 2025

•

5 pages

Key Methods for Factorizing Algebraic Expressions

Layla Guthrie

@laylaguthrie_aoly

Factorization is a key algebra skill that helps simplify complex expressions into products of simpler ones. Understanding different factorization methods allows you to solve equations more efficiently and recognize patterns in algebraic expressions.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Methods of Factorization: Finding the GCF

Finding the Greatest Common Factor (GCF) is your first step in factoring expressions. It's like identifying what several terms share in common.

To find the GCF of numbers, break them down into their prime factors. For example, with 24 and 36, write 24 = 2×2×2×3 and 36 = 2×2×3×3. The common factors (2×2×3 = 12) give us the GCF.

For algebraic expressions like 7x³, 14x², and 21x⁴, find the GCF in two parts:

Find the GCF of the coefficients (7 in this case)
Find the variable with the lowest power (x² here) So the GCF is 7x².

Pro Tip: Always look for the GCF first before trying other factoring methods - it makes the next steps much easier!

When expressions contain binomials like 3x $a+b$ and 2y $a+b$ , the common factor is simply $a+b$ . Remember that your goal is to pull out everything that's common to all terms.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Factoring Out the GCF in Complex Expressions

Factoring gets more interesting with longer expressions. The key is to identify what's common to all terms and pull it out.

With expressions like 15y³+12y⁴, start by finding the GCF - here it's 3y³. Then rewrite as 3y³ $5+4y$ . For 9a⁴b-18a⁵b+27ab, the GCF is 9ab, giving us 9ab $1-2a+3a²$ .

When you see expressions like 2w $x+3$ -5 $x+3$ , look for the common binomial factor. Here $x+3$ appears in both terms, so we can factor it out: $x+3$ $2w-5$ .

Remember: Sometimes you need to factor multiple times. In 16w⁴-40w³-12w²+30w, first factor out 2w to get 2w $8w³-20w²-6w+15$ , then look for other patterns in what remains.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Factoring Trinomials and Special Products

The A.C. method is super helpful for factoring trinomials like ax²+bx+c:

Find the product of a and c (that's ac)
Look for two numbers that multiply to give ac and add up to b
Split the middle term using these numbers
Group and factor

For example, with 2x²+7x+6:

ac = 12
We need numbers that multiply to 12 and add to 7 (that's 3 and 4)
Rewrite as 2x²+4x+3x+6
Factor by grouping: $2x+3$ $x+2$

Difference of squares follows the pattern a²-b²= $a+b$ $a-b$ . So y²-25 = $y+5$ $y-5$ . But remember: sum of squares $like p²+9$ cannot be factored with real numbers.

Watch out! Don't try to factor sum of squares like a²+b² - they're prime over real numbers and can't be factored further!

For complex expressions like w⁴-81, recognize that it's (w²)²-9² and factor it as $w²+9$ $w²-9$ , then factor w²-9 again to get $w²+9$ $w+3$ $w-3$ .

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Perfect Square Trinomials and Advanced Factoring

Perfect square trinomials follow these patterns:

a²+2ab+b² = $a+b$ ²
a²-2ab+b² = $a-b$ ²

To spot them, check if:

The first and last terms are perfect squares
The middle term equals 2ab or -2ab

For example, x²+14x+49 is a perfect square trinomial because:

First term is x² (perfect square)
Last term is 49 = 7² (perfect square)
Middle term is 14x, which equals 2(x)(7)
Therefore, x²+14x+49 = $x+7$ ²

Another example: 25y²-20y+4 = $5y-2$ ² because:

25y² = (5y)²
4 = 2²
-20y = -2(5y)(2)

When factoring complex expressions like 18c³-48c²d+32cd², look for a common factor first. Here we have 2c, so we get 2c $9c²-24cd+16d²$ , which reveals a perfect square trinomial: 2c $3c-4d$ ².

Smart approach: Always check if your trinomial might be a perfect square - it makes factoring much faster than using other methods!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Sum and Difference of Cubes

The sum of cubes follows the pattern: a³+b³ = $a+b$ $a²-ab+b²$

The difference of cubes follows: a³-b³ = $a-b$ $a²+ab+b²$

These patterns are super useful for expressions like w³+64:

Recognize that 64 = 4³
Apply the formula: w³+4³ = $w+4$ $w²-4w+16$

For more complex examples like 27p³-1000q³:

Rewrite as (3p)³-(10q)³
Apply the difference of cubes formula: $3p-10q$ $9p²+30pq+100q²$

These factoring techniques work together. For 3y⁴-48:

Factor out common factor: 3 $y⁴-16$
Recognize y⁴-16 as a difference of squares: $y²-4$ $y²+4$
Factor y²-4 as another difference of squares: $y-2$ $y+2$
Final answer: 3 $y-2$ $y+2$ $y²+4$

Factoring strategy: When you see expressions with cubes or higher powers, think about using these special patterns - they save a ton of work compared to other methods!

We thought you’d never ask...

What is the Knowunity AI companion?

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Smart Tools NEW

Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines

Mock Exam

Quiz

Flashcards

Essay

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user