Question

Use long division or synthetic division to divide the given polynomial.

(x^3-17x+32)/(x+5)




A
x2+5x−8−8x+5x^{2}+5x-8-\frac{8}{x+5}x
2
+5x−8−
x+5
8
​

B
x2−5x+8−8x+5x^{2}-5x+8-\frac{8}{x+5}x
2
−5x+8−
x+5
8
​

C
x2+5x−8x^{2}+5x-8x
2
+5x−8
D
x2−10x+2−18x+5x^{2}-10x+2-\frac{18}{x+5}x
2
−10x+2−
x+5
18
​

Answer 1

To divide the polynomial \( x^3 - 17x + 32 \) by \( x + 5 \) using long division, follow these steps:

1. Set up the long division:

- Divide \( x^3 \) by \( x \), which gives \( x^2 \).
- Multiply \( x^2 \) by \( x + 5 \) to get \( x^3 + 5x^2 \).
- Subtract this from the original polynomial:

\[
(x^3 - 17x + 32) - (x^3 + 5x^2) = -5x^2 - 17x + 32
\]

2. Next, bring down the terms:

- Divide \( -5x^2 \) by \( x \), which gives \( -5x \).
- Multiply \( -5x \) by \( x + 5 \) to get \( -5x^2 - 25x \).
- Subtract this from the current polynomial:

\[
(-5x^2 - 17x + 32) - (-5x^2 - 25x) = 8x + 32
\]

3. Next, continue with the division:

- Divide \( 8x \) by \( x \) to get \( 8 \).
- Multiply \( 8 \) by \( x + 5 \) to get \( 8x + 40 \).
- Subtract this from the current polynomial:

\[
(8x + 32) - (8x + 40) = -8
\]

4. Now, we have our results:

The quotient is \( x^2 - 5x + 8 \) and the remainder is \(-8\).

This means:

\[
\frac{x^3 - 17x + 32}{x + 5} = x^2 - 5x + 8 - \frac{8}{x + 5}
\]

Thus, the correct answer is **B: \( x^2 - 5x + 8 - \frac{8}{x + 5} \)**.