Match each polynomial expression to its completely factored form.

$x^2 - 16$ $(x - 4)(x + 4)$
$x^2 + 6x + 8$ $(x + 2)(x + 4)$
$2x^2 - 12x + 18$ $2(x - 3)^2$
$3x^2 - 5x - 2$ $(3x + 1)(x - 2)$

Answer

The correct matches pair each polynomial with its mathematically equivalent fully factored expression.

Each polynomial requires identifying its specific factoring archetype: the difference of two squares, a standard quadratic trinomial, GCF extraction followed by recognizing a perfect square trinomial, or factoring a non-monic trinomial.

Step-by-Step Solution

Analyze the binomial

x^2 - 16

Matches to

(x - 4)(x + 4)

It represents a difference of perfect squares.

Analyze the trinomial

x^2 + 6x + 8

Matches to

(x + 2)(x + 4)

The constants 2 and 4 have a product of 8 and a sum of 6.

Analyze the polynomial

2x^2 - 12x + 18

Matches to

2(x - 3)^2

Factoring out the common factor of 2 leaves

x^2 - 6x + 9

, which is the square of

(x - 3)

Analyze the polynomial

3x^2 - 5x - 2

Matches to

(3x + 1)(x - 2)

Multiplying the binomials

(3x + 1)

and

(x - 2)

confirms it expands to the original expression.

Key Concept

Factoring different archetypes of polynomials, including the difference of squares, monic trinomials, non-monic trinomials, and extracting the Greatest Common Factor (GCF).

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