Question

Use long division to divide the polynomial 35x9−42x8−5x5+6x4 by 5x2−6x . Write your answer in standard form.(1 point)

Answer 1

To divide the polynomial \( 35x^9 - 42x^8 - 5x^5 + 6x^4 \) by \( 5x^2 - 6x \) using long division, we'll follow the steps systematically.

1. **Start with the dividend and divisor**:
\[
\text{Dividend} = 35x^9 - 42x^8 - 5x^5 + 6x^4
\]
\[
\text{Divisor} = 5x^2 - 6x
\]

2. **Divide the leading term of the dividend by the leading term of the divisor**:
\[
\frac{35x^9}{5x^2} = 7x^7
\]

3. **Multiply the entire divisor by \( 7x^7 \)**:
\[
7x^7(5x^2 - 6x) = 35x^9 - 42x^8
\]

4. **Subtract this from the original dividend**:
\[
(35x^9 - 42x^8 - 5x^5 + 6x^4) - (35x^9 - 42x^8) = 0 - 5x^5 + 6x^4
\]
So, we have:
\[
-5x^5 + 6x^4
\]

5. **Repeat the division**. Now we have:
\[
\text{New Dividend} = -5x^5 + 6x^4
\]
Divide the leading term:
\[
\frac{-5x^5}{5x^2} = -x^3
\]

6. **Multiply the entire divisor by \(-x^3\)**:
\[
-x^3(5x^2 - 6x) = -5x^5 + 6x^4
\]

7. **Subtract this from the current dividend**:
\[
(-5x^5 + 6x^4) - (-5x^5 + 6x^4) = 0
\]

8. **Now combine the results**:
The result from the long division is:
\[
7x^7 - x^3
\]

Thus, the final answer, written in standard form, is:
\[
\boxed{7x^7 - x^3}
\]