Which of the following expressions represents the complete factorization of 2x^4 - 32? — ACT

Which of the following expressions represents the complete factorization of $2x^4 - 32$ ?

A
$2(x^2 - 4)(x^2 + 4)$
B
$(2x^2 - 8)(x^2 + 4)$
C
$2(x - 2)(x + 2)(x + 2)^2$
$2(x - 2)(x + 2)(x^2 + 4)$ Cevap
E
$2(x - 4)(x + 4)$

2(x - 2)(x + 2)(x^2 + 4)

The correct answer is found by first factoring out the greatest common factor of

2

, resulting in

2(x^4 - 16)

. Next, recognize that

x^4 - 16

is a difference of squares,

(x^2)^2 - 4^2

, which factors into

(x^2 - 4)(x^2 + 4)

. Finally, factor the remaining difference of squares,

x^2 - 4

, into

(x - 2)(x + 2)

. The sum of squares,

x^2 + 4

, cannot be factored further. Combining these parts gives the complete factorization.

Factor out the greatest common factor from the polynomial.

2(x^4 - 16)

Both terms of

2x^4 - 32

are divisible by

2

, so we factor it out to simplify the remaining expression.

Factor the difference of squares inside the parentheses.

2(x^2 - 4)(x^2 + 4)

The expression

x^4 - 16

is a difference of squares because it can be written as

(x^2)^2 - 4^2

Factor the remaining difference of squares.

2(x - 2)(x + 2)(x^2 + 4)

The binomial

x^2 - 4

is also a difference of squares (

x^2 - 2^2

), which factors into

(x - 2)(x + 2)

. The sum of squares

x^2 + 4

cannot be factored further using real numbers.

Factoring polynomials completely by extracting the greatest common factor and repeatedly applying the difference of squares formula.

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