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 \(x + 5\), we can use synthetic division or long division. Here, I will use polynomial long division.

**Step 1: Setup the division.**
Divide \(x^3 - 17x + 32\) by \(x + 5\).

```
____________________
x + 5 | x^3 + 0x^2 - 17x + 32
```

**Step 2: Divide the leading terms.**
The leading term of \(x^3\) divided by the leading term of \(x\) (in \(x + 5\)) gives \(x^2\):

```
x^2
____________________
x + 5 | x^3 + 0x^2 - 17x + 32
- (x^3 + 5x^2)
________________
-5x^2 - 17x
```

**Step 3: Bring down the next term.**
Now we have \(-5x^2 - 17x\).

**Step 4: Repeat the process.**
Divide \(-5x^2\) by \(x\) to get \(-5x\):

```
x^2 - 5x
____________________
x + 5 | x^3 + 0x^2 - 17x + 32
- (x^3 + 5x^2)
________________
-5x^2 - 17x
- (-5x^2 - 25x)
________________
8x + 32
```

**Step 5: Bring down the next term.**
Now we have \(8x + 32\).

**Step 6: Repeat again.**
Now divide \(8x\) by \(x\) to get \(8\):

```
x^2 - 5x + 8
____________________
x + 5 | x^3 + 0x^2 - 17x + 32
- (x^3 + 5x^2)
________________
-5x^2 - 17x
- (-5x^2 - 25x)
________________
8x + 32
- (8x + 40)
___________
-8
```

**Step 7: Write the result.**
The quotient is \(x^2 - 5x + 8\) and the remainder is \(-8\).

So, the complete result of the division is:

\[
x^2 - 5x + 8 - \frac{8}{x + 5}
\]

From the options given, the correct answer corresponds with:

**B**: \(x^2 - 5x + 8 - \frac{8}{x + 5}\)