Which of the following is a factor of the polynomial 8x^3 - 12x^2 - 18x + 27? — ACT

Which of the following is a factor of the polynomial $8x^3 - 12x^2 - 18x + 27$ ?

The correct answer is

2x + 3

The correct answer is the factor

2x + 3

. By grouping the first two terms and the last two terms, we get

4x^2(2x - 3) - 9(2x - 3)

. Factoring out the common binomial yields

(4x^2 - 9)(2x - 3)

. The term

4x^2 - 9

is a difference of squares, which factors into

(2x - 3)(2x + 3)

. Therefore, the fully factored expression is

(2x - 3)^2(2x + 3)

, making the linear expression with a positive constant term the correct factor.

Group the four terms of the polynomial into two pairs.

(8x^3 - 12x^2) - (18x - 27)

Grouping terms allows us to find common binomial factors in a cubic polynomial.

Factor out the greatest common factor (GCF) from each group.

4x^2(2x - 3) - 9(2x - 3)

The GCF of

8x^3

and

12x^2

4x^2

, and the GCF of

-18x

and

27

-9

(factoring out the negative to match the binomials).

Factor out the common binomial factor

(2x - 3)

(4x^2 - 9)(2x - 3)

Both groups share the factor

(2x - 3)

, so it can be factored out.

Factor the quadratic difference of squares

(4x^2 - 9)

(2x - 3)(2x + 3)(2x - 3) = (2x - 3)^2(2x + 3)

The term

4x^2 - 9

is a difference of squares of the form

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

, where

a = 2x

and

b = 3

Factoring a cubic polynomial completely by grouping and then applying the difference of squares formula.

Soru