Question

Divide these polynomials: (8x4−12x3+12x2−4x+7)÷(4x2−2x).

Answer 1

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

1. **Set Up the Division:**
- Dividend: \( 8x^4 - 12x^3 + 12x^2 - 4x + 7 \)
- Divisor: \( 4x^2 - 2x \)

2. **Divide the Leading Terms:**
- Divide \( 8x^4 \) by \( 4x^2 \) to get \( 2x^2 \).

3. **Multiply the Divisor by the Result:**
- \( 2x^2(4x^2 - 2x) = 8x^4 - 4x^3 \).

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

5. **Repeat the Process:**
- Now, divide the leading term \( -8x^3 \) by \( 4x^2 \) to get \( -2x \).
- Multiply the divisor by \( -2x \):
\[
-2x(4x^2 - 2x) = -8x^3 + 4x^2
\]
- Subtract from the current polynomial:
\[
(-8x^3 + 12x^2 - 4x + 7) - (-8x^3 + 4x^2) = (12x^2 - 4x^2) - 4x + 7 = 8x^2 - 4x + 7
\]

6. **Continue Dividing:**
- Divide \( 8x^2 \) by \( 4x^2 \) to get \( 2 \).
- Multiply the divisor by \( 2 \):
\[
2(4x^2 - 2x) = 8x^2 - 4x
\]
- Subtract from the current polynomial:
\[
(8x^2 - 4x + 7) - (8x^2 - 4x) = 7
\]

Now, we have reached the point where the remainder is less in degree than the divisor.

7. **Final Result:**
The division yields:
\[
2x^2 - 2 + \frac{7}{4x^2 - 2x}
\]

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