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

9-6 Factoring Quadratic Trinomials

In this Grade 9 lesson from California Reveal Math, Algebra 1, students learn to factor quadratic trinomials of the form ax² + bx + c by identifying two integers whose sum equals b and whose product equals c. The lesson covers cases where the leading coefficient a equals 1 with both positive and negative constant terms, and introduces the concept of a prime polynomial when no integer factor pairs exist. Students practice using factor tables and algebra tiles to write trinomials in factored form as a product of two binomials.

Section 1

Factoring quadratic trinomials

Property

To factor $x^2 + bx + c$ , we look for two numbers $p$ and $q$ so that

$pq = c$ and $p + q = b$

When we expand the factored form $(x + p)(x + q)$ , we get $x^2 + (p + q)x + pq$ .

Section 2

Sign patterns for factoring

Property

Assume that $b$ , $c$ , $p$ , and $q$ are positive integers.

$x^2 + bx + c = (x + p)(x + q)$ . If all the coefficients of the trinomial are positive, then both $p$ and $q$ are positive.
$x^2 - bx + c = (x - p)(x - q)$ . If the linear term of the trinomial is negative and the other two terms positive, then $p$ and $q$ are both negative.
$x^2 \pm bx - c = (x + p)(x - q)$ . If the constant term of the trinomial is negative, then $p$ and $q$ have opposite signs.

Examples

To factor $x^2 - 7x + 10$ , the constant term is positive and the middle term is negative. So, we need two negative numbers that multiply to 10 and add to -7. The factors are $(x - 2)(x - 5)$ .
In $y^2 + 4y - 12$ , the constant term is negative, so the factors have opposite signs. We need numbers that multiply to -12 and add to 4. The factorization is $(y + 6)(y - 2)$ .
For $z^2 - z - 30$ , the negative constant term means opposite signs. We need numbers that multiply to -30 and add to -1. The correct pair is -6 and 5, so we get $(z - 6)(z + 5)$ .

Explanation

The signs in the trinomial give you huge clues! A positive last term means the signs in your factors are the same. A negative last term means the signs are different. Use this to factor faster!

Section 3

Factoring Out the GCF Before Factoring a Trinomial

Property

Before factoring a quadratic trinomial $ax^2 + bx + c$ , always check whether all terms share a common factor. If a Greatest Common Factor (GCF) exists, factor it out first:

ax^2 + bx + c = \text{GCF} \cdot \left(\frac{a}{\text{GCF}}x^2 + \frac{b}{\text{GCF}}x + \frac{c}{\text{GCF}}\right)

Book overview

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Unit 9: Polynomials

Back to California Reveal Math, Algebra 1

Lesson 1
9-1 Adding and Subtracting Polynomials
Learn Practice
Lesson 2
9-2 Multiplying Polynomials by Monomials
Learn Practice
Lesson 3
9-3 Multiplying Polynomials
Learn Practice
Lesson 4
9-4 Special Products
Learn Practice
Lesson 5
9-5 Factoring Polynomials
Learn Practice
Lesson 6Current
9-6 Factoring Quadratic Trinomials
Learn Practice
Lesson 7
9-7 Factoring Special Products
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Factoring quadratic trinomials

Property

To factor $x^2 + bx + c$ , we look for two numbers $p$ and $q$ so that

$pq = c$ and $p + q = b$

When we expand the factored form $(x + p)(x + q)$ , we get $x^2 + (p + q)x + pq$ .

Section 2

Sign patterns for factoring

Property

Assume that $b$ , $c$ , $p$ , and $q$ are positive integers.

$x^2 + bx + c = (x + p)(x + q)$ . If all the coefficients of the trinomial are positive, then both $p$ and $q$ are positive.
$x^2 - bx + c = (x - p)(x - q)$ . If the linear term of the trinomial is negative and the other two terms positive, then $p$ and $q$ are both negative.
$x^2 \pm bx - c = (x + p)(x - q)$ . If the constant term of the trinomial is negative, then $p$ and $q$ have opposite signs.

Examples

To factor $x^2 - 7x + 10$ , the constant term is positive and the middle term is negative. So, we need two negative numbers that multiply to 10 and add to -7. The factors are $(x - 2)(x - 5)$ .
In $y^2 + 4y - 12$ , the constant term is negative, so the factors have opposite signs. We need numbers that multiply to -12 and add to 4. The factorization is $(y + 6)(y - 2)$ .
For $z^2 - z - 30$ , the negative constant term means opposite signs. We need numbers that multiply to -30 and add to -1. The correct pair is -6 and 5, so we get $(z - 6)(z + 5)$ .

Explanation

The signs in the trinomial give you huge clues! A positive last term means the signs in your factors are the same. A negative last term means the signs are different. Use this to factor faster!

Section 3

Factoring Out the GCF Before Factoring a Trinomial

Property

Before factoring a quadratic trinomial $ax^2 + bx + c$ , always check whether all terms share a common factor. If a Greatest Common Factor (GCF) exists, factor it out first:

ax^2 + bx + c = \text{GCF} \cdot \left(\frac{a}{\text{GCF}}x^2 + \frac{b}{\text{GCF}}x + \frac{c}{\text{GCF}}\right)

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Unit 9: Polynomials

Back to California Reveal Math, Algebra 1

Lesson 1
9-1 Adding and Subtracting Polynomials
Learn Practice
Lesson 2
9-2 Multiplying Polynomials by Monomials
Learn Practice
Lesson 3
9-3 Multiplying Polynomials
Learn Practice
Lesson 4
9-4 Special Products
Learn Practice
Lesson 5
9-5 Factoring Polynomials
Learn Practice
Lesson 6Current
9-6 Factoring Quadratic Trinomials
Learn Practice
Lesson 7
9-7 Factoring Special Products
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

Factoring quadratic trinomials

Property

To factor $x^2 + bx + c$ , we look for two numbers $p$ and $q$ so that

$pq = c$ and $p + q = b$

When we expand the factored form $(x + p)(x + q)$ , we get $x^2 + (p + q)x + pq$ .

Section 2

Sign patterns for factoring

Property

Assume that $b$ , $c$ , $p$ , and $q$ are positive integers.

$x^2 + bx + c = (x + p)(x + q)$ . If all the coefficients of the trinomial are positive, then both $p$ and $q$ are positive.
$x^2 - bx + c = (x - p)(x - q)$ . If the linear term of the trinomial is negative and the other two terms positive, then $p$ and $q$ are both negative.
$x^2 \pm bx - c = (x + p)(x - q)$ . If the constant term of the trinomial is negative, then $p$ and $q$ have opposite signs.

Examples

To factor $x^2 - 7x + 10$ , the constant term is positive and the middle term is negative. So, we need two negative numbers that multiply to 10 and add to -7. The factors are $(x - 2)(x - 5)$ .
In $y^2 + 4y - 12$ , the constant term is negative, so the factors have opposite signs. We need numbers that multiply to -12 and add to 4. The factorization is $(y + 6)(y - 2)$ .
For $z^2 - z - 30$ , the negative constant term means opposite signs. We need numbers that multiply to -30 and add to -1. The correct pair is -6 and 5, so we get $(z - 6)(z + 5)$ .

Explanation

The signs in the trinomial give you huge clues! A positive last term means the signs in your factors are the same. A negative last term means the signs are different. Use this to factor faster!

Section 3

Factoring Out the GCF Before Factoring a Trinomial

Property

Before factoring a quadratic trinomial $ax^2 + bx + c$ , always check whether all terms share a common factor. If a Greatest Common Factor (GCF) exists, factor it out first:

ax^2 + bx + c = \text{GCF} \cdot \left(\frac{a}{\text{GCF}}x^2 + \frac{b}{\text{GCF}}x + \frac{c}{\text{GCF}}\right)

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Unit 9: Polynomials

Back to California Reveal Math, Algebra 1

Lesson 1
9-1 Adding and Subtracting Polynomials
Learn Practice
Lesson 2
9-2 Multiplying Polynomials by Monomials
Learn Practice
Lesson 3
9-3 Multiplying Polynomials
Learn Practice
Lesson 4
9-4 Special Products
Learn Practice
Lesson 5
9-5 Factoring Polynomials
Learn Practice
Lesson 6Current
9-6 Factoring Quadratic Trinomials
Learn Practice
Lesson 7
9-7 Factoring Special Products
Learn Practice

9-6 Factoring Quadratic Trinomials

Property

Property

Examples

Explanation

Property

Unit 9: Polynomials

9-1 Adding and Subtracting Polynomials

9-2 Multiplying Polynomials by Monomials

9-3 Multiplying Polynomials

9-4 Special Products

9-5 Factoring Polynomials

9-6 Factoring Quadratic Trinomials

9-7 Factoring Special Products

Lesson overview

Property

Property

Examples

Explanation

Property

Unit 9: Polynomials

9-1 Adding and Subtracting Polynomials

9-2 Multiplying Polynomials by Monomials

9-3 Multiplying Polynomials

9-4 Special Products

9-5 Factoring Polynomials

9-6 Factoring Quadratic Trinomials

9-7 Factoring Special Products

9-1 Adding and Subtracting Polynomials

9-2 Multiplying Polynomials by Monomials

9-3 Multiplying Polynomials

9-4 Special Products

9-5 Factoring Polynomials

9-6 Factoring Quadratic Trinomials

9-7 Factoring Special Products