Question

Divide the third-degree polynomial, 8x3+20x2−12x , by the first-degree polynomial, x+3 , using long division. What is the result?

Answer 1

To divide the polynomial \( 8x^3 + 20x^2 - 12x \) by \( x + 3 \) using long division, we follow these steps:

1. **Set up the long division**:
\[
\begin{array}{r|rrrr}
& 8x^2 - 4x + 0 \\
\hline
x + 3 & 8x^3 + 20x^2 - 12x \\
\end{array}
\]

2. **Divide the leading term**:
- Divide the leading term of the dividend \( 8x^3 \) by the leading term of the divisor \( x \):
\[
\frac{8x^3}{x} = 8x^2
\]

3. **Multiply and subtract**:
- Multiply \( 8x^2 \) by \( x + 3 \) and subtract from the original polynomial:
\[
(8x^2)(x + 3) = 8x^3 + 24x^2
\]
- Subtract this from \( 8x^3 + 20x^2 - 12x \):
\[
(8x^3 + 20x^2 - 12x) - (8x^3 + 24x^2) = -4x^2 - 12x
\]

4. **Repeat the process**:
- Now, divide \( -4x^2 \) by \( x \):
\[
\frac{-4x^2}{x} = -4x
\]
- Multiply \( -4x \) by \( x + 3 \):
\[
(-4x)(x + 3) = -4x^2 - 12x
\]
- Subtract this from \( -4x^2 - 12x \):
\[
(-4x^2 - 12x) - (-4x^2 - 12x) = 0
\]

5. **Combine results**:
- The dividend has been completely divided with no remainder. Thus, the result of the division is:

\[
\text{Result} = 8x^2 - 4x
\]

Therefore, the quotient when \( 8x^3 + 20x^2 - 12x \) is divided by \( x + 3 \) is \( \boxed{8x^2 - 4x} \).