Question

Use long division to divide the polynomial −30x6+48x5+50x−80%0D%0A−%0D%0A30%0D%0A𝑥%0D%0A6%0D%0A+%0D%0A48%0D%0A𝑥%0D%0A5%0D%0A+%0D%0A50%0D%0A𝑥%0D%0A−%0D%0A80%0D%0A by 5x−8%0D%0A5%0D%0A𝑥%0D%0A−%0D%0A8%0D%0A. Write your answer in standard form.(1 point)

Answer 1

To divide the polynomial \(-30x^6 + 48x^5 + 50x - 80\) by \(5x - 8\) using long division, follow these steps:

1. **Set Up the Division:**
Write the polynomial \(-30x^6 + 48x^5 + 0x^4 + 0x^3 + 50x - 80\) (note the placeholders for missing degrees) under the long division bar, and write \(5x - 8\) outside.

2. **Divide the Leading Terms:**
Divide the leading term of the dividend by the leading term of the divisor.

\[
\frac{-30x^6}{5x} = -6x^5
\]

3. **Multiply and Subtract:**
Multiply \(-6x^5\) by \(5x - 8\) and subtract from the original polynomial.

\[
-6x^5(5x - 8) = -30x^6 + 48x^5
\]

Subtract this from the original polynomial:

\[
(-30x^6 + 48x^5 + 0x^4 + 0x^3 + 50x - 80) - (-30x^6 + 48x^5) = 0x^6 + 0x^5 + 0x^4 + 0x^3 + 50x - 80
\]
This simplifies to:

\[
0 + 0 + 0 + 50x - 80
\]

4. **Repeat the Process:**
Now divide the new leading term by \(5x\):

\[
\frac{50x}{5x} = 10
\]

Multiply \(10\) by \(5x - 8\):

\[
10(5x - 8) = 50x - 80
\]

Subtract this from the polynomial:

\[
(50x - 80) - (50x - 80) = 0
\]

5. **Conclusion:**
The remainder is \(0\), and the final result of the division is:

\[
-6x^5 + 10
\]

Thus, the answer in standard form is:

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