Which of the following is the completely factored form of x^2 - 64? — ACT

Which of the following is the completely factored form of $x^2 - 64$ ?

A
$(x - 8)^2$
B
$(x - 32)(x + 32)$
$(x - 8)(x + 8)$ Answer
D
$(x^2 - 8)(x^2 + 8)$
E
$(x - 16)(x + 4)$

(x - 8)(x + 8)

The polynomial

x^2 - 64

is a difference of two squares because

x^2

is the square of

x

and

64

is the square of

8

. Applying the difference of squares formula,

a^2 - b^2 = (a - b)(a + b)

, with

a = x

and

b = 8

gives the factored form

(x - 8)(x + 8)

Identify the structure of the polynomial.

The expression

x^2 - 64

is a difference of two squares since

x^2

(x)^2

and

64

(8)^2

Recognizing the difference of squares allows the use of the algebraic identity

a^2 - b^2 = (a - b)(a + b)

Apply the difference of squares identity with

a = x

and

b = 8

(x - 8)(x + 8)

Substituting

x

and

8

into the formula yields the factored form.

Factoring the difference of squares using the identity

a^2 - b^2 = (a - b)(a + b)

Question