Question

13
select the correct answer.
one factor of the polynomial $2x^3 - 3x^2 - 3x + 2$ is $(x - 2)$. which expression represents the other factor, or factors, of the polynomial?
a. $(2x^2 - x + 1)$
b. $(2x + 1)(x - 1)$
c. $(2x - 1)(x + 1)$
d. $(2x^2 + 1)$

Explanation:

Step1: Set up polynomial division

We divide $2x^3 - 3x^2 - 3x + 2$ by $x - 2$ using polynomial long division.

Step2: Divide leading terms

Divide $2x^3$ by $x$ to get $2x^2$. Multiply $x-2$ by $2x^2$: $2x^2(x-2)=2x^3-4x^2$.
Subtract from the original polynomial:
$$(2x^3 - 3x^2 - 3x + 2) - (2x^3 - 4x^2) = x^2 - 3x + 2$$

Step3: Divide new leading terms

Divide $x^2$ by $x$ to get $x$. Multiply $x-2$ by $x$: $x(x-2)=x^2-2x$.
Subtract:
$$(x^2 - 3x + 2) - (x^2 - 2x) = -x + 2$$

Step4: Divide final leading terms

Divide $-x$ by $x$ to get $-1$. Multiply $x-2$ by $-1$: $-1(x-2)=-x+2$.
Subtract:
$$(-x + 2) - (-x + 2) = 0$$

Step5: Verify the quotient

The quotient from division is $2x^2 + x - 1$. Factor this quadratic:
$$2x^2 + x - 1 = (2x - 1)(x + 1)$$

Answer:

C. $(2x - 1)(x + 1)$