Pair the given polynomial expressions on the left with their fully factored equivalents on… — Armed Services Vocational Aptitude Battery (ASVAB)

Pair the given polynomial expressions on the left with their fully factored equivalents on the right.

$4x^2 - 25$ $(2x - 5)(2x + 5)$
$x^2 - 8x + 15$ $(x - 3)(x - 5)$
$3x^2 + 12x - 15$ $3(x + 5)(x - 1)$
$2x^3 - 8x$ $2x(x - 2)(x + 2)$

Answer

The correct pairings are:

4x^2 - 25

with

(2x - 5)(2x + 5)

;

x^2 - 8x + 15

with

(x - 3)(x - 5)

;

3x^2 + 12x - 15

with

3(x + 5)(x - 1)

; and

2x^3 - 8x

with

2x(x - 2)(x + 2)

Each polynomial expression was broken down into its fundamental algebraic components by identifying patterns such as the difference of squares, standard trinomial factoring, and extracting the greatest common factor.

Step-by-Step Solution

Analyze the first expression

4x^2 - 25

It factors to

(2x - 5)(2x + 5)

It follows the difference of squares pattern

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

with

a = 2x

and

b = 5

Analyze the second expression

x^2 - 8x + 15

It factors to

(x - 3)(x - 5)

The constant term is positive and the middle coefficient is negative, so both factors must contain negative constants that multiply to

15

and add to

-8

Analyze the third expression

3x^2 + 12x - 15

It factors to

3(x + 5)(x - 1)

Factoring out the GCF of

3

gives

3(x^2 + 4x - 5)

. The resulting quadratic factors into binomials with constants

5

and

-1

Analyze the fourth expression

2x^3 - 8x

It factors to

2x(x - 2)(x + 2)

Factoring out the GCF of

2x

gives

2x(x^2 - 4)

. The expression

x^2 - 4

is a difference of squares that further factors into

(x - 2)(x + 2)

Key Concept

Factoring Polynomials

Estimated Time:2m 0s

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