Question

Divide these polynomials: (8x^4−12x^3+12x^2−4x+7)÷(4x2−2x) .(1 point)

Answer 1

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

1. **Setup the division**: Write the division in long division format.
\[
\begin{array}{r|l}
4x^2 - 2x & 8x^4 - 12x^3 + 12x^2 - 4x + 7
\end{array}
\]

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

3. **Multiply and subtract**: Multiply \( 2x^2 \) by \( (4x^2 - 2x) \) and subtract it from the original polynomial.
\[
2x^2 \cdot (4x^2 - 2x) = 8x^4 - 4x^3
\]
Subtract:
\[
(8x^4 - 12x^3) - (8x^4 - 4x^3) = -8x^3
\]

4. **Bring down the next term**: Bring down the next term from the original polynomial:
\[
-8x^3 + 12x^2
\]

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

6. **Multiply and subtract**: Multiply \( -2x \) by \( (4x^2 - 2x) \):
\[
-2x \cdot (4x^2 - 2x) = -8x^3 + 4x^2
\]
Subtract:
\[
(-8x^3 + 12x^2) - (-8x^3 + 4x^2) = 8x^2
\]

7. **Bring down the next term**: Bring down the next term:
\[
8x^2 - 4x
\]

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

9. **Multiply and subtract**: Multiply \( 2 \) by \( (4x^2 - 2x) \):
\[
2 \cdot (4x^2 - 2x) = 8x^2 - 4x
\]
Subtract:
\[
(8x^2 - 4x) - (8x^2 - 4x) = 0
\]

10. **Result**: Since the remainder is 0, the division completes successfully. The quotient is:
\[
2x^2 - 2x + 2
\]

Thus, the result of dividing \( (8x^4 - 12x^3 + 12x^2 - 4x + 7) \) by \( (4x^2 - 2x) \) is:
\[
2x^2 - 2x + 2 + \frac{7}{4x^2 - 2x}
\]

So the full answer is:
\[
2x^2 - 2x + 2 + \frac{7}{4x^2 - 2x}
\]