Question

division with polynomials: practice
p(x) = 3x³ + 11x² - 4x - 6
which statements are true?
□ (x + 2) is a factor of p.
□ p(-2) = 22
□ the quotient of p(x) and (x + 2) is 3x² + 5x + 6 + 6/(x + 2).
□ the quotient of p(x) and (x + 2) is 3x² + 5x - 14 + 22/(x + 2).

Explanation:

To determine which statements are true, we use the Factor Theorem and polynomial long division (or synthetic division). The Factor Theorem states that \((x - a)\) is a factor of \(p(x)\) if and only if \(p(a) = 0\). For \((x + 2)\), this means \(a = -2\), so we check \(p(-2)\).

Step 1: Evaluate \(p(-2)\)

Given \(p(x) = 3x^3 + 11x^2 - 4x - 6\), substitute \(x = -2\):
\[

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

\]

Step 2: Analyze \((x + 2)\) as a factor

By the Factor Theorem, \((x + 2)\) is a factor of \(p(x)\) only if \(p(-2) = 0\). But \(p(-2) = 22
eq 0\), so \((x + 2)\) is not a factor of \(p(x)\).

Step 3: Divide \(p(x)\) by \((x + 2)\) (Polynomial Long Division)

Divide \(3x^3 + 11x^2 - 4x - 6\) by \(x + 2\):

1. Divide \(3x^3\) by \(x\) to get \(3x^2\). Multiply \(x + 2\) by \(3x^2\): \(3x^3 + 6x^2\). Subtract from \(p(x)\):

\[
(3x^3 + 11x^2 - 4x - 6) - (3x^3 + 6x^2) = 5x^2 - 4x - 6
\]

1. Divide \(5x^2\) by \(x\) to get \(5x\). Multiply \(x + 2\) by \(5x\): \(5x^2 + 10x\). Subtract:

\[
(5x^2 - 4x - 6) - (5x^2 + 10x) = -14x - 6
\]

1. Divide \(-14x\) by \(x\) to get \(-14\). Multiply \(x + 2\) by \(-14\): \(-14x - 28\). Subtract:

\[
(-14x - 6) - (-14x - 28) = 22
\]

Thus, the quotient is \(3x^2 + 5x - 14\) with a remainder of \(22\). So:
\[
\frac{p(x)}{x + 2} = 3x^2 + 5x - 14 + \frac{22}{x + 2}
\]

Evaluate Each Statement:

• \((x + 2)\) is a factor of \(p\): False (since \(p(-2)

eq 0\)).

• \(p(-2) = 22\): True (from Step 1).
• Quotient: \(3x^2 + 5x + 6 + \frac{6}{x + 2}\): False (does not match the division result).
• Quotient: \(3x^2 + 5x - 14 + \frac{22}{x + 2}\): True (matches the division result).

Answer:

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

(Note: The first statement is false, and the third is false. The second and fourth are true.)