Question

consider this polynomial. ( p(x) = 3x^3 + 11x^2 - 4x - 6 ) which statements are true? the quotient of ( p(x) ) and ( (x + 2) ) is ( 3x^2 + 5x + 6 + \frac{6}{x + 2} ). ( p(-2) = 22 ) ( p(-2) = 6 ) ( (x + 2) ) is a factor of ( p ). the quotient of ( p(x) ) and ( (x + 2) ) is ( 3x^2 + 5x - 14 + \frac{22}{x + 2} ).

Explanation:

Step1: Calculate $p(-2)$

Substitute $x=-2$ into $p(x)$:

$$\begin{align*} p(-2)&=3(-2)^3 + 11(-2)^2 - 4(-2) - 6\\ &=3(-8)+11(4)+8-6\\ &=-24+44+8-6\\ &=22 \end{align*}$$

Step2: Verify factor theorem

For $(x+2)$ to be a factor, $p(-2)$ must equal 0. Since $p(-2)=22
eq0$, $(x+2)$ is not a factor.

Step3: Polynomial long division

Divide $p(x)=3x^3+11x^2-4x-6$ by $(x+2)$:

1. First term: $\frac{3x^3}{x}=3x^2$, multiply $(x+2)$ by $3x^2$: $3x^3+6x^2$
2. Subtract from $p(x)$: $(3x^3+11x^2-4x-6)-(3x^3+6x^2)=5x^2-4x-6$
3. Next term: $\frac{5x^2}{x}=5x$, multiply $(x+2)$ by $5x$: $5x^2+10x$
4. Subtract: $(5x^2-4x-6)-(5x^2+10x)=-14x-6$
5. Next term: $\frac{-14x}{x}=-14$, multiply $(x+2)$ by $-14$: $-14x-28$
6. Subtract: $(-14x-6)-(-14x-28)=22$

So the quotient is $3x^2+5x-14+\frac{22}{x+2}$.

Answer:

• $p(-2) = 22$
• The quotient of $p(x)$ and $(x + 2)$ is $3x^2 + 5x - 14 + \frac{22}{x+2}$.