Question

Use long division to divide the polynomial −30x^6+48x^5+50x−80 by 5x−8 . Write your answer in standard form

Answer 1

To perform long division on the polynomial \(-30x^6 + 48x^5 + 50x - 80\) divided by \(5x - 8\), we will follow the long division process step by step.

1. **Set up the division**: Write \( -30x^6 + 48x^5 + 0x^4 + 50x - 80\) inside the division symbol and \(5x - 8\) outside.

2. **Divide the leading term**: Divide \(-30x^6\) by \(5x\):
\[
\frac{-30x^6}{5x} = -6x^5
\]
Write \(-6x^5\) above the division bar.

3. **Multiply**: Multiply \(-6x^5\) by \(5x - 8\):
\[
-6x^5(5x - 8) = -30x^6 + 48x^5
\]

4. **Subtract**: Subtract this product from the original polynomial:
\[
(-30x^6 + 48x^5 + 0x^4 + 50x - 80) - (-30x^6 + 48x^5) = 0x^6 + 0x^5 + 0x^4 + 50x - 80 = 50x - 80
\]

5. **Bring down**: Since there are no more higher degree terms to bring down, we proceed with \(50x - 80\).

6. **Repeat the process**: Divide \(50x\) by \(5x\):
\[
\frac{50x}{5x} = 10
\]
Write \(10\) above the division bar.

7. **Multiply**: Multiply \(10\) by \(5x - 8\):
\[
10(5x - 8) = 50x - 80
\]

8. **Subtract**: Subtract this product from \(50x - 80\):
\[
(50x - 80) - (50x - 80) = 0
\]

9. **Result**: Since there is no remainder, the division is complete.

Collecting the results, we have:

\[
-30x^6 + 48x^5 + 50x - 80 \div (5x - 8) = -6x^5 + 10
\]

Thus, the final answer in standard form is:

\[
\boxed{-6x^5 + 10}
\]