Learn on PengiCalifornia Reveal Math, Algebra 1Unit 9: Polynomials

9-5 Factoring Polynomials

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to factor polynomials using the Distributive Property and factoring by grouping. The lesson covers finding the greatest common factor (GCF) of polynomial terms and applying it to rewrite expressions in factored form, as well as grouping four-term polynomials to identify common binomial factors. Real-world applications, such as modeling projectile height with expressions like 440t − 16t², reinforce how factoring relates to multiplying polynomials.

Section 1

Factoring a Polynomial

Property

Factoring a polynomial is the inverse of the Distributive Property. It rewrites a sum or difference of monomials as a product of factors.

Examples

$6x^3 + 8x^2 - 2x = 2x(3x^2 + 4x - 1)$
$9x^4y^2 - 9x^5y = 9x^4y(y - x^2)$

Explanation

Think of this as 'un-distributing'! You find the GCF of all the terms, pull it out to the front, and write what’s left over inside parentheses. It's a great way to simplify complex expressions.

Section 2

Factoring the Greatest Common Factor

Property

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials. To factor out the GCF:

Identify the GCF of the coefficients.
Identify the GCF of the variables (using the lowest power for each variable).
Combine to find the expression's GCF.
Determine what the GCF must be multiplied by to get each term.
Write the factored expression as the GCF multiplied by the remaining terms.

Examples

To factor $8a^2b^3 + 12ab^2 + 20a^3b$ , the GCF is $4ab$ . Factoring this out gives $4ab(2ab^2 + 3b + 5a^2)$ .
To factor $15m^4n - 25m^2n^2 + 5mn$ , the GCF is $5mn$ . The factored form is $5mn(3m^3 - 5mn + 1)$ .
To factor $12x^2(y-1) + 8x(y-1)$ , the GCF is $4x(y-1)$ . This results in $4x(y-1)(3x + 2)$ .

Explanation

Think of this as reverse distribution. You are finding the largest number and variable part that every single term shares and pulling it out to the front. This simplifies the expression inside the parentheses, making it easier to work with.

Section 3

Factor by Grouping

Property

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

HOW TO: Factor by grouping.
Step 1. Group terms with common factors.
Step 2. Factor out the common factor in each group.
Step 3. Factor the common factor from the expression.
Step 4. Check by multiplying the factors.

Examples

Factor $ab + 5a + 3b + 15$ . Group the terms: $(ab + 5a) + (3b + 15)$ . Factor the GCF from each group: $a(b+5) + 3(b+5)$ . Factor out the common binomial: $(b+5)(a+3)$ .
Factor $x^2 + 2x - 5x - 10$ . Group the terms: $(x^2 + 2x) + (-5x - 10)$ . Factor GCFs: $x(x+2) - 5(x+2)$ . Factor out the common binomial: $(x+2)(x-5)$ .
Factor $mn - 8m + 4n - 32$ . Group the terms: $(mn - 8m) + (4n - 32)$ . Factor GCFs: $m(n-8) + 4(n-8)$ . Factor out the common binomial: $(n-8)(m+4)$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Factoring a Polynomial

Property

Factoring a polynomial is the inverse of the Distributive Property. It rewrites a sum or difference of monomials as a product of factors.

Examples

$6x^3 + 8x^2 - 2x = 2x(3x^2 + 4x - 1)$
$9x^4y^2 - 9x^5y = 9x^4y(y - x^2)$

Explanation

Think of this as 'un-distributing'! You find the GCF of all the terms, pull it out to the front, and write what’s left over inside parentheses. It's a great way to simplify complex expressions.

Section 2

Factoring the Greatest Common Factor

Property

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials. To factor out the GCF:

Identify the GCF of the coefficients.
Identify the GCF of the variables (using the lowest power for each variable).
Combine to find the expression's GCF.
Determine what the GCF must be multiplied by to get each term.
Write the factored expression as the GCF multiplied by the remaining terms.

Examples

To factor $8a^2b^3 + 12ab^2 + 20a^3b$ , the GCF is $4ab$ . Factoring this out gives $4ab(2ab^2 + 3b + 5a^2)$ .
To factor $15m^4n - 25m^2n^2 + 5mn$ , the GCF is $5mn$ . The factored form is $5mn(3m^3 - 5mn + 1)$ .
To factor $12x^2(y-1) + 8x(y-1)$ , the GCF is $4x(y-1)$ . This results in $4x(y-1)(3x + 2)$ .

Explanation

Section 3

Factor by Grouping

Property

Examples

Factor $ab + 5a + 3b + 15$ . Group the terms: $(ab + 5a) + (3b + 15)$ . Factor the GCF from each group: $a(b+5) + 3(b+5)$ . Factor out the common binomial: $(b+5)(a+3)$ .
Factor $x^2 + 2x - 5x - 10$ . Group the terms: $(x^2 + 2x) + (-5x - 10)$ . Factor GCFs: $x(x+2) - 5(x+2)$ . Factor out the common binomial: $(x+2)(x-5)$ .
Factor $mn - 8m + 4n - 32$ . Group the terms: $(mn - 8m) + (4n - 32)$ . Factor GCFs: $m(n-8) + 4(n-8)$ . Factor out the common binomial: $(n-8)(m+4)$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Factoring a Polynomial

Property

Factoring a polynomial is the inverse of the Distributive Property. It rewrites a sum or difference of monomials as a product of factors.

Examples

$6x^3 + 8x^2 - 2x = 2x(3x^2 + 4x - 1)$
$9x^4y^2 - 9x^5y = 9x^4y(y - x^2)$

Explanation

Think of this as 'un-distributing'! You find the GCF of all the terms, pull it out to the front, and write what’s left over inside parentheses. It's a great way to simplify complex expressions.

Section 2

Factoring the Greatest Common Factor

Property

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials. To factor out the GCF:

Identify the GCF of the coefficients.
Identify the GCF of the variables (using the lowest power for each variable).
Combine to find the expression's GCF.
Determine what the GCF must be multiplied by to get each term.
Write the factored expression as the GCF multiplied by the remaining terms.

Examples

To factor $8a^2b^3 + 12ab^2 + 20a^3b$ , the GCF is $4ab$ . Factoring this out gives $4ab(2ab^2 + 3b + 5a^2)$ .
To factor $15m^4n - 25m^2n^2 + 5mn$ , the GCF is $5mn$ . The factored form is $5mn(3m^3 - 5mn + 1)$ .
To factor $12x^2(y-1) + 8x(y-1)$ , the GCF is $4x(y-1)$ . This results in $4x(y-1)(3x + 2)$ .

Explanation

Section 3

Factor by Grouping

Property

Examples

Factor $ab + 5a + 3b + 15$ . Group the terms: $(ab + 5a) + (3b + 15)$ . Factor the GCF from each group: $a(b+5) + 3(b+5)$ . Factor out the common binomial: $(b+5)(a+3)$ .
Factor $x^2 + 2x - 5x - 10$ . Group the terms: $(x^2 + 2x) + (-5x - 10)$ . Factor GCFs: $x(x+2) - 5(x+2)$ . Factor out the common binomial: $(x+2)(x-5)$ .
Factor $mn - 8m + 4n - 32$ . Group the terms: $(mn - 8m) + (4n - 32)$ . Factor GCFs: $m(n-8) + 4(n-8)$ . Factor out the common binomial: $(n-8)(m+4)$ .