Use division of polynomials to find the quotient and remainder when:
P(x) = x^3 + 2x^2 - 17x + 6 is divided by D(x) = x - 3.

To find the quotient and remainder when dividing the polynomial \( P(x) = x^3 + 2x^2 - 17x + 6 \) by \( D(x) = x - 3 \), we'll use polynomial division. Here's the step-by-step process:

### Step 1: Set up the division
We divide \( x^3 + 2x^2 - 17x + 6 \) by \( x - 3 \).

### Step 2: Divide the leading term
- The leading term of \( P(x) \) is \( x^3 \).
- The leading term of \( D(x) \) is \( x \).

Divide \( x^3 \) by \( x \) to get \( x^2 \).

### Step 3: Multiply and subtract
- Multiply \( x^2 \) by \( D(x) = x - 3 \) to get \( x^3 - 3x^2 \).
- Subtract \( x^3 - 3x^2 \) from \( P(x) = x^3 + 2x^2 - 17x + 6 \):

\[
(x^3 + 2x^2 - 17x + 6) - (x^3 - 3x^2) = 5x^2 - 17x + 6
\]

### Step 4: Repeat the process
- Divide the new leading term \( 5x^2 \) by \( x \) to get \( 5x \).
- Multiply \( 5x \) by \( x - 3 \) to get \( 5x^2 - 15x \).
- Subtract \( 5x^2 - 15x \) from \( 5x^2 - 17x + 6 \):

\[
(5x^2 - 17x + 6) - (5x^2 - 15x) = -2x + 6
\]

### Step 5: Repeat the process again
- Divide the new leading term \( -2x \) by \( x \) to get \( -2 \).
- Multiply \( -2 \) by \( x - 3 \) to get \( -2x + 6 \).
- Subtract \( -2x + 6 \) from \( -2x + 6 \):

\[
(-2x + 6) - (-2x + 6) = 0
\]

### Step 6: Write the final answer
Since the remainder is 0, the division is exact.

The **quotient** is \( Q(x) = x^2 + 5x - 2 \) and the **remainder** is \( R(x) = 0 \).

### Summary:
\[
\frac{x^3 + 2x^2 - 17x + 6}{x - 3} = x^2 + 5x - 2 \text{ with a remainder of } 0
\]

Would you like any further details or have any questions?

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Here are 8 related questions you might find helpful:
1. How do you perform synthetic division for polynomials?
2. What is the Remainder Theorem and how does it apply here?
3. Can you divide polynomials with higher degrees using the same method?
4. How do you find the roots of a polynomial using division?
5. What is the relationship between the quotient and the factors of the polynomial?
6. How does synthetic division differ from long division in polynomials?
7. Can this method be applied to polynomials with complex coefficients?
8. How does polynomial division relate to the Factor Theorem?

**Tip:** Always verify your quotient and remainder by multiplying the divisor with the quotient and adding the remainder to check if you get back the original polynomial.

Learn how to use polynomial division to find the quotient and remainder when dividing P(x) = x^3 + 2x^2 - 17x + 6 by D(x) = x - 3. Step-by-step solution included. Suitable for Grades 10-12 students.

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