Question

Use long division to divide the polynomial 33x^5 + 22x^4 - 50x^3 - 26^2 + 13x by 3x^2 + 2x - 1

Answer 1

To divide the polynomial \(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x\) by \(3x^2 + 2x - 1\) using long division, we proceed as follows:

1. **Setup**: Write the dividend \(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x\) under the long division symbol and the divisor \(3x^2 + 2x - 1\) outside.

2. **First Division**: Divide the leading term of the dividend \(33x^5\) by the leading term of the divisor \(3x^2\):
\[
\frac{33x^5}{3x^2} = 11x^3
\]
Write \(11x^3\) above the long division symbol.

3. **Multiply**: Multiply \(11x^3\) by the entire divisor \(3x^2 + 2x - 1\):
\[
11x^3(3x^2 + 2x - 1) = 33x^5 + 22x^4 - 11x^3
\]

4. **Subtract**: Subtract this result from the original polynomial:
\[
(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x) - (33x^5 + 22x^4 - 11x^3)
\]
This simplifies to:
\[
(-50x^3 + 11x^3) - 26x^2 + 13x = -39x^3 - 26x^2 + 13x
\]

5. **Next Division**: Now divide the leading term \(-39x^3\) by \(3x^2\):
\[
\frac{-39x^3}{3x^2} = -13x
\]
Write \(-13x\) above the long division symbol next to \(11x^3\).

6. **Multiply**: Multiply \(-13x\) by the divisor:
\[
-13x(3x^2 + 2x - 1) = -39x^3 - 26x^2 + 13x
\]

7. **Subtract**: Subtract this from the current polynomial:
\[
(-39x^3 - 26x^2 + 13x) - (-39x^3 - 26x^2 + 13x) = 0
\]

At this point, the polynomial has been divided completely since the remainder is 0.

Thus, the result of the long division is:
\[
\text{Quotient} = 11x^3 - 13x
\]

So, the quotient when dividing \(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x\) by \(3x^2 + 2x - 1\) is:
\[
\boxed{11x^3 - 13x}
\]