Question

divide these polynomials (8x^4-12x^3+12x^2-4x+7) divided by (4x^2-2x)

Answer 1

To divide the polynomial \( 8x^4 - 12x^3 + 12x^2 - 4x + 7 \) by \( 4x^2 - 2x \), we can use polynomial long division. Here’s how the division proceeds:

1. **Divide the leading terms**: Divide the leading term of the dividend (\( 8x^4 \)) by the leading term of the divisor (\( 4x^2 \)):
\[
\frac{8x^4}{4x^2} = 2x^2
\]

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

3. **Subtract**: Subtract this result from the original polynomial:
\[
(8x^4 - 12x^3 + 12x^2 - 4x + 7) - (8x^4 - 4x^3) = (-12x^3 + 4x^3 + 12x^2 - 4x + 7) = -8x^3 + 12x^2 - 4x + 7
\]

4. **Repeat**: Now repeat the process with the new polynomial \(-8x^3 + 12x^2 - 4x + 7\).

- **Divide the leading terms**:
\[
\frac{-8x^3}{4x^2} = -2x
\]

- **Multiply**:
\[
-2x(4x^2 - 2x) = -8x^3 + 4x^2
\]

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

5. **Continue**: Continue with \( 8x^2 - 4x + 7 \).

- **Divide the leading terms**:
\[
\frac{8x^2}{4x^2} = 2
\]

- **Multiply**:
\[
2(4x^2 - 2x) = 8x^2 - 4x
\]

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

Now, we have reached a remainder of \( 7 \).

### Final Result

The result of the division is:
\[
2x^2 - 2x + 2 \quad \text{with a remainder of } 7
\]
Thus, the final answer is:
\[
\frac{8x^4 - 12x^3 + 12x^2 - 4x + 7}{4x^2 - 2x} = 2x^2 - 2x + 2 + \frac{7}{4x^2 - 2x}
\]