p(x) ≡ 2x3 − 9x2 − 2x + 11. a Find the remainder when p(x) is divided by (x + 2). (2) b Find the quotient and remainder when p(x) is divided by (x − 4).

To solve this, let’s address each part separately.

### Part (a): Find the remainder when \( p(x) \) is divided by \( (x + 2) \).

To find the remainder when a polynomial \( p(x) \) is divided by \( x + a \), we can use the **Remainder Theorem**. According to the theorem, the remainder of the division of \( p(x) \) by \( x + 2 \) is \( p(-2) \).

Given:
\[
p(x) = 2x^3 - 9x^2 - 2x + 11
\]
we substitute \( x = -2 \):

\[
p(-2) = 2(-2)^3 - 9(-2)^2 - 2(-2) + 11
\]

Calculating each term:

1. \( 2(-2)^3 = 2 \cdot (-8) = -16 \)
2. \( -9(-2)^2 = -9 \cdot 4 = -36 \)
3. \( -2(-2) = 4 \)
4. \( + 11 = 11 \)

Adding these terms:

\[
p(-2) = -16 - 36 + 4 + 11 = -37
\]

Thus, the remainder when \( p(x) \) is divided by \( x + 2 \) is **-37**.

### Part (b): Find the quotient and remainder when \( p(x) \) is divided by \( (x - 4) \).

To find the quotient and remainder, we can use **synthetic division** or **long division**. Here, I’ll perform synthetic division with \( p(x) \) and \( x - 4 \) (where \( x = 4 \)).

Given \( p(x) = 2x^3 - 9x^2 - 2x + 11 \), let’s set up synthetic division with \( x = 4 \).

1. Write down the coefficients of \( p(x) \): \( 2, -9, -2, 11 \).
2. Perform synthetic division:

\[
\begin{array}{r|rrrr}
4 & 2 & -9 & -2 & 11 \\
   &   & 8 & -4 & -24 \\
\hline
   & 2 & -1 & -6 & -13 \\
\end{array}
\]

In this division:
- The last value, \(-13\), is the **remainder**.
- The other values form the **quotient**, represented as \( 2x^2 - x - 6 \).

So, the quotient is \( 2x^2 - x - 6 \) and the remainder is **-13**.

### Summary of Answers
- **Part (a)**: Remainder = -37.
- **Part (b)**: Quotient = \( 2x^2 - x - 6 \), Remainder = -13.

Would you like more details on these steps?

#### Related Questions:
1. How does synthetic division differ from polynomial long division?
2. What is the Remainder Theorem and how is it applied in polynomial division?
3. How can we verify the quotient and remainder for polynomial division?
4. Can the Remainder Theorem be used for any polynomial and divisor?
5. How is the Factor Theorem related to the Remainder Theorem?

#### Tip:
Remember that the Remainder Theorem is a quick way to find remainders without needing to fully divide the polynomial!

p(x) ≡ 2x^3 − 9x^2 − 2x + 11. a) Find the remainder when p(x) is divided by (x + 2). b) Find the quotient and remainder when p(x) is divided by (x − 4).

Explore the step-by-step solution for finding the remainder when p(x) = 2x^3 - 9x^2 - 2x + 11 is divided by (x + 2) and the quotient and remainder when divided by (x - 4). Covers key concepts like the Remainder Theorem and Synthetic Division. Ideal for high school students in Grades 10-12.

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