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 will proceed through the standard long division steps.

1. **Set up the division.**

We want to divide \(35x^9 - 42x^8 - 5x^5 + 6x^4\) (the dividend) by \(5x^2 - 6x\) (the divisor).

2. **Divide the leading term.**

Divide the leading term of the dividend \(35x^9\) by the leading term of the divisor \(5x^2\):

\[
\frac{35x^9}{5x^2} = 7x^7
\]

3. **Multiply and subtract.**

Multiply the entire divisor \(5x^2 - 6x\) by \(7x^7\):

\[
7x^7 \cdot (5x^2 - 6x) = 35x^9 - 42x^8
\]

Now subtract this from the original polynomial:

\[
(35x^9 - 42x^8 - 5x^5 + 6x^4) - (35x^9 - 42x^8) = -5x^5 + 6x^4
\]

4. **Repeat the process.**

Now divide the leading term \(-5x^5\) by \(5x^2\):

\[
\frac{-5x^5}{5x^2} = -x^3
\]

Multiply the divisor \(5x^2 - 6x\) by \(-x^3\):

\[
-x^3 \cdot (5x^2 - 6x) = -5x^5 + 6x^4
\]

Subtract this from \(-5x^5 + 6x^4\):

\[
(-5x^5 + 6x^4) - (-5x^5 + 6x^4) = 0
\]

Since we reached 0, the division has completed.

5. **Result.**

The quotient from the long division is:

\[
7x^7 - x^3
\]

Therefore, the final answer in standard form is:

\[
\boxed{7x^7 - x^3}
\]