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Unpacking the Standards
Understanding the standards and the vocabulary terms in the standards will
help you know exactly what you are expected to learn in this chapter.
CC.9-12.A.SSE.2
What It Means For You
Lessons 7-1, 7-2, 7-3, 7-4, 7-5, 7-6
Use the structure of an expression to
identify ways to rewrite it.
You will learn to factor expressions, which means you will rewrite
them as a product of two or more expressions. Being able to recognize
patterns will help you decide which method to use.
Key Vocabulary
EXAMPLE
expression (expresión) A
mathematical phrase that contains
operations, numbers, and/or
variables.
Factor x
​ 2​ ​ + 7x + 6
The algebra tiles below show that ​x2​ ​+ 7x + 6 = (x + 1)(x + 6).
x + 6
x + 1
Factor ​6x​2​ - 11x + 3
EXAMPLE
6​x2​ ​- 11x + 3
6x ⋅ x or 3x ⋅ 2x
1 ⋅ 3
Guess and check: (6x - 1)(x - 3) = 6​x2​ ​- 18x - x + 3
= 6​x​2​- 19x + 3 ✗
Guess and check: (3x - 1)(2x - 3) = 6​x2​ ​- 9x - 2x + 3
= 6​x​2​- 11x + 3 ✓
EXAMPLE
Factor x
​ 2​ ​ - 49
Use the difference of two squares pattern:
Chapter 7
​a2​ ​- ​b2​ ​= (a + b)(a - b)
x​ ​2​- 49 = (x + 7)(x - 7)
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Factoring Polynomials
© Houghton Mifflin Harcourt Publishing Company
chapter 7
×
What It Means For You
CC.9-12.A.SSE.1b
Interpret complicated expressions by
viewing one or more of their parts as a
single entity.
Key Vocabulary
expression (expresión) A
mathematical phrase that contains
operations, numbers, and/or
variables.
Lessons 7-1, 7-2, 7-5, 7-6
You will learn to recognize when the terms of an expression share
common factors. You will also learn to recognize when an expression
fits a certain pattern based on the structure of its parts.
EXAMPLE
Factor Out the Greatest Common Factor
8​x3​ ​​y​2​ + 6​x2​ ​y + 10xy =
2xy(4​x2​ ​y) + 2xy(3x) + 2xy(5) =
2xy(4​x​2​y + 3x + 5)
EXAMPLE
Factor by Grouping
3​x2​ ​ + 6x + 5x + 10 =
3x(x) + 3x(2) + 5(x) + 5(2) =
3x(x + 2) + 5(x + 2) =
(3x + 5)(x + 2)
EXAMPLE
Factor a Perfect-Square Trinomial
A square quilt is made up of squares sewn together in a grid pattern
with a one-foot border on all sides. The side length of each square is
x feet. The area in square feet is given by the expression
100​x2​ ​+ 40x + 4. Write an expression for the side length of the quilt.
Use a formula: Area of square = Side length squared
100​x ​2​+ 40x + 4 = ​(10x + 2)​2​
Factor:
The side length in feet is given by the expression 10x + 2.
Chapter 7
369
Factoring Polynomials
chapter 7
© Houghton Mifflin Harcourt Publishing Company; Photo credit: © Ruth Black/Alamy Images
Key Vocabulary
binomial (binomio) A polynomial with two terms.
difference of two squares (diferencia de dos cuadrados) A polynomial of the form ​a2​ ​- ​b2​ ​,
which may be written as the product (a + b)(a - b).
expression (expresión) A mathematical phrase that contains operations, numbers, and/or variables.
factor (factor) A number or expression that is multiplied by another number or expression to get a
product. See also factoring.
factoring (factorización) The process of writing a number or algebraic expression as a product.
greatest common factor (monomials) (GCF) (máximo común divisor (monomios)
(MCD)) The product of the greatest integer and the greatest power of each variable that divide evenly into
each monomial.
monomial (monomio) A number or a product of numbers and variables with whole-number exponents,
or a polynomial with one term.
chapter 7
perfect-square trinomial (trinomio cuadrado perfecto) A trinomial whose factored form is the
square of a binomial. A perfect-square trinomial has the form a
​ ​2​- 2ab + ​b2​ ​= ​(a - b)​ 2​or
2
2
​a​ ​+ 2ab + b
​ ​ ​= ​(a + b)​ 2​.
polynomial (polinomio) A monomial or a sum or difference of monomials.
trinomial (trinomio) A polynomial with three terms.
The Common Core Standards for Mathematical Practice describe varieties
of expertise that mathematics educators at all levels should seek to develop
in their students. Opportunities to develop these practices are integrated
throughout this program.
1. M
ake sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. C
onstruct viable arguments and critique the reasoning of others.
4. M
odel with mathematics.
5. Use appropriate tools strategically.
6. A
ttend to precision.
7. L ook for and make use of structure.
8. L ook for and express regularity in repeated reasoning
Chapter 7
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Factoring Polynomials