Which of the following is the completely factored form of the expression 9x^2 - 36? — ACT

Which of the following is the completely factored form of the expression $9x^2 - 36$ ?

A
$(3x - 6)^2$
B
$(3x - 6)(3x + 6)$
$9(x - 2)(x + 2)$ Cevap
D
$9(x^2 - 4)$
E
$9(x - 2)^2$

The correct factored form is

9(x - 2)(x + 2)

The expression

9x^2 - 36

has a greatest common factor of

9

. Factoring out

9

yields

9(x^2 - 4)

. The term inside the parentheses,

x^2 - 4

, is a difference of squares that can be factored as

(x - 2)(x + 2)

. Combining these gives the completely factored form

9(x - 2)(x + 2)

Identify and factor out the greatest common factor (GCF) of the terms in the expression.

The GCF of

9x^2

and

36

9

. Factoring it out gives

9(x^2 - 4)

Factoring out the GCF simplifies the remaining polynomial and is the first step in completely factoring an expression.

Recognize and factor the quadratic expression inside the parentheses.

The term

x^2 - 4

is a difference of squares of the form

a^2 - b^2

, where

a = x

and

b = 2

. Factoring it yields

(x - 2)(x + 2)

A difference of squares

a^2 - b^2

always factors into

(a - b)(a + b)

Combine the factored terms to write the completely factored expression.

Combining the GCF and the factored binomials gives

9(x - 2)(x + 2)

This represents the expression written as a product of its prime polynomial factors.

Factoring polynomials by first extracting the greatest common factor (GCF) and then applying the difference of squares identity.

Instead of factoring out the greatest common factor first, the expression can be factored as a difference of squares directly:

9x^2 - 36 = (3x)^2 - 6^2 = (3x - 6)(3x + 6)

. Then, a common factor of

3

can be factored out from each binomial:

3(x - 2) \cdot 3(x + 2) = 9(x - 2)(x + 2)

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