Question

What is the quotient of (5x4−10x3−13x2+12x+30)÷(x2−4x+4) ?(1 point) Responses 5x2+10x+7 5 x squared plus 10 x plus 7 5x4−10x3−14x2+16x+26 5 x superscript 4 baseline minus 10 x cubed minus 14 x squared plus 16 x plus 26 5x3+10x2+7x+2 5 x cubed plus 10 x squared plus 7 x plus 2 5x2+10x+7+2x2−4x+4

Answer 1

To find the quotient of \((5x^4 - 10x^3 - 13x^2 + 12x + 30) \div (x^2 - 4x + 4)\), we can use polynomial long division.

1. **Set up the division** by writing \(5x^4 - 10x^3 - 13x^2 + 12x + 30\) under the long division and \(x^2 - 4x + 4\) outside.

2. **Divide the leading terms**:
- Divide \(5x^4\) by \(x^2\) to get \(5x^2\).

3. **Multiply** \(5x^2\) by \(x^2 - 4x + 4\):
\[
5x^2(x^2 - 4x + 4) = 5x^4 - 20x^3 + 20x^2
\]

4. **Subtract** this from the original polynomial:
\[
(5x^4 - 10x^3 - 13x^2 + 12x + 30) - (5x^4 - 20x^3 + 20x^2)
\]
This simplifies to:
\[
(0 + 10x^3 - 33x^2 + 12x + 30)
\]

5. **Repeat** the process. Divide \(10x^3\) by \(x^2\) to get \(10x\).

6. **Multiply** \(10x\) by \(x^2 - 4x + 4\):
\[
10x(x^2 - 4x + 4) = 10x^3 - 40x^2 + 40x
\]

7. **Subtract** this:
\[
(10x^3 - 33x^2 + 12x + 30) - (10x^3 - 40x^2 + 40x)
\]
This simplifies to:
\[
(0 + 7x^2 - 28x + 30)
\]

8. **Repeat** again. Divide \(7x^2\) by \(x^2\) to get \(7\).

9. **Multiply** \(7\) by \(x^2 - 4x + 4\):
\[
7(x^2 - 4x + 4) = 7x^2 - 28x + 28
\]

10. **Subtract**:
\[
(7x^2 - 28x + 30) - (7x^2 - 28x + 28)
\]
This simplifies to:
\[
0 + 0 + 2 = 2
\]

Putting this all together yields:
\[
\frac{5x^4 - 10x^3 - 13x^2 + 12x + 30}{x^2 - 4x + 4} = 5x^2 + 10x + 7 + \frac{2}{x^2 - 4x + 4}
\]

From the given responses, the quotient we computed, without the remainder, matches:
**5x² + 10x + 7**.

Thus, the final answer is: **5x² + 10x + 7**.