Question

long division to divide the polynomials: x3+11x2+15x/x+3 (1 point) Responses x2+8x+9+−27x+3 x squared plus 8 x plus 9 plus the fraction with numerator negative 27 and denominator x plus 3 x2+8x−9+27x+3 x squared plus 8 x minus 9 plus the fraction with numerator 27 and denominator x plus 3 x2+8x−9 x squared plus 8 x minus 9 x2+8x+9

Answer 1

To divide the polynomial \( x^3 + 11x^2 + 15x \) by \( x + 3 \) using long division, follow these steps:

1. **Setup the Long Division**:
\[
\begin{array}{r|l}
x + 3 & x^3 + 11x^2 + 15x \\
\end{array}
\]

2. **Divide the leading term**:
Divide \( x^3 \) by \( x \) to get \( x^2 \).

3. **Multiply**:
Multiply \( x^2 \) by \( x + 3 \):
\[
x^2(x + 3) = x^3 + 3x^2
\]

4. **Subtract**:
Subtract this from the original polynomial:
\[
(x^3 + 11x^2 + 15x) - (x^3 + 3x^2) = 8x^2 + 15x
\]

5. **Repeat the process**:
Now, divide \( 8x^2 \) by \( x \) to get \( 8x \).

6. **Multiply**:
Multiply \( 8x \) by \( x + 3 \):
\[
8x(x + 3) = 8x^2 + 24x
\]

7. **Subtract**:
Subtract this from \( 8x^2 + 15x \):
\[
(8x^2 + 15x) - (8x^2 + 24x) = -9x
\]

8. **Continue**:
Now, divide \( -9x \) by \( x \) to get \( -9 \).

9. **Multiply**:
Multiply \( -9 \) by \( x + 3 \):
\[
-9(x + 3) = -9x - 27
\]

10. **Subtract**:
Subtract from \( -9x \):
\[
(-9x) - (-9x - 27) = 27
\]

Putting all this together, we find:
\[
\frac{x^3 + 11x^2 + 15x}{x + 3} = x^2 + 8x - 9 + \frac{27}{x + 3}
\]

So the result is:
\[
x^2 + 8x - 9 + \frac{27}{x + 3}
\]

The correct response from your options would be:
\[
x^2 + 8x - 9 + \frac{27}{x + 3}
\]