Question

select the correct answer. one factor of the polynomial ( 6x^3 - x^2 + 8x + 5 ) is ( (2x + 1) ). what is the other factor of the polynomial? (note: use long division.)
a. ( (3x^2 + 5) )
b. ( (3x^2 - 2) )
c. ( (3x^2 - 2x + 5) )
d. ( (3x^2 + 5x - 2) )

Explanation:

Step1: Divide leading terms

Divide $6x^3$ by $2x$: $\frac{6x^3}{2x}=3x^2$

Step2: Multiply divisor by result

Multiply $(2x+1)$ by $3x^2$: $(2x+1) \cdot 3x^2 = 6x^3 + 3x^2$

Step3: Subtract from dividend

Subtract from $6x^3 - x^2 + 8x + 5$:
$$(6x^3 - x^2 + 8x + 5) - (6x^3 + 3x^2) = -4x^2 + 8x + 5$$

Step4: Divide new leading terms

Divide $-4x^2$ by $2x$: $\frac{-4x^2}{2x}=-2x$

Step5: Multiply divisor by result

Multiply $(2x+1)$ by $-2x$: $(2x+1) \cdot (-2x) = -4x^2 - 2x$

Step6: Subtract from new polynomial

Subtract from $-4x^2 + 8x + 5$:
$$(-4x^2 + 8x + 5) - (-4x^2 - 2x) = 10x + 5$$

Step7: Divide leading terms again

Divide $10x$ by $2x$: $\frac{10x}{2x}=5$

Step8: Multiply divisor by result

Multiply $(2x+1)$ by $5$: $(2x+1) \cdot 5 = 10x + 5$

Step9: Subtract to find remainder

Subtract from $10x + 5$:
$$(10x + 5) - (10x + 5) = 0$$
Combine the terms from Steps1,4,7: $3x^2 - 2x + 5$

Answer:

C. $(3x^2 - 2x + 5)$