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.

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

Cevap

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.

Adım Adım Çözüm

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)

Anahtar Kavram

Factoring Polynomials

Tahmini Süre:2m 0s

Soru

Cevap

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