Question

Divide these polynomials: (8x4−12x3+12x2−4x+7)÷(4x2−2x) .(1 point) Responses 2x3−3x2+3x−1+74x2−2x 2 x cubed minus 3 x squared plus 3 x minus 1 plus Start Fraction 7 over 4 x squared minus 2 x End Fraction 2x2−2x+2+74x2−2x 2 x squared minus 2 x plus 2 plus Start Fraction 7 over 4 x squared minus 2 x End Fraction 2x2−2x+2 2 x squared minus 2 x plus 2 2x3−2x2+2x+7

Answer 1

To divide the polynomials \( (8x^4 - 12x^3 + 12x^2 - 4x + 7) \div (4x^2 - 2x) \), we can use polynomial long division.

1. **Set up the division**:
\[
\text{Dividend: } 8x^4 - 12x^3 + 12x^2 - 4x + 7
\]
\[
\text{Divisor: } 4x^2 - 2x
\]

2. **Divide the leading terms**:
Divide \( 8x^4 \) by \( 4x^2 \) to get \( 2x^2 \).

3. **Multiply** \( 2x^2 \) by the entire divisor:
\[
2x^2(4x^2 - 2x) = 8x^4 - 4x^3
\]

4. **Subtract**:
\[
(8x^4 - 12x^3 + 12x^2 - 4x + 7) - (8x^4 - 4x^3) = -8x^3 + 12x^2 - 4x + 7
\]

5. **Repeat the process**:
Divide \( -8x^3 \) by \( 4x^2 \) to get \( -2x \).

6. **Multiply** \( -2x \) by the divisor:
\[
-2x(4x^2 - 2x) = -8x^3 + 4x^2
\]

7. **Subtract**:
\[
(-8x^3 + 12x^2 - 4x + 7) - (-8x^3 + 4x^2) = 8x^2 - 4x + 7
\]

8. **Repeat again**:
Divide \( 8x^2 \) by \( 4x^2 \) to get \( 2 \).

9. **Multiply** \( 2 \) by the divisor:
\[
2(4x^2 - 2x) = 8x^2 - 4x
\]

10. **Subtract**:
\[
(8x^2 - 4x + 7) - (8x^2 - 4x) = 7
\]

At this point, we cannot divide anymore because the degree of the remainder \( 7 \) is less than the degree of the divisor \( 4x^2 - 2x \).

Thus, the result of the polynomial long division is:

\[
2x^2 - 2x + 2 + \frac{7}{4x^2 - 2x}
\]

Therefore, the correct answer is:
**\( 2x^2 - 2x + 2 + \frac{7}{4x^2 - 2x} \)**.