Question

Use long division to divide the polynomial 33x^5+22x^4−50x3−26x^2+13x by 3x^2+2x−1 . Write your answer in standard form.

Answer 1

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

1. **Set Up the Long Division**: We place the dividing polynomial \( 3x^2 + 2x - 1 \) on the outside, and the dividend \( 33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x \) on the inside.

2. **Divide the Leading Terms**: Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{33x^5}{3x^2} = 11x^3
\]
We write \( 11x^3 \) above the division line.

3. **Multiply and Subtract**: Multiply \( 11x^3 \) by the entire divisor \( 3x^2 + 2x - 1 \):
\[
11x^3(3x^2 + 2x - 1) = 33x^5 + 22x^4 - 11x^3
\]
Now, we subtract this from the original polynomial:
\[
(33x^5 + 22x^4 - 50x^3) - (33x^5 + 22x^4 - 11x^3) = (-50x^3 + 11x^3) - 26x^2 + 13x = -39x^3 - 26x^2 + 13x
\]

4. **Repeat the Process**: Now, take the new polynomial \( -39x^3 - 26x^2 + 13x \) and divide the leading term by the leading term of the divisor:
\[
\frac{-39x^3}{3x^2} = -13x
\]
Write \( -13x \) above the division line.

5. **Multiply and Subtract Again**:
\[
-13x(3x^2 + 2x - 1) = -39x^3 - 26x^2 + 13x
\]
Subtract this from the current polynomial:
\[
(-39x^3 - 26x^2 + 13x) - (-39x^3 - 26x^2 + 13x) = 0
\]

6. **Continue the Process**: At this stage, we have no more terms to bring down, but we can confirm if there are any leftover terms from higher-degree dividend terms. Since all terms matched perfectly, we check if we have lowered polynomials to divide further.

Thus, we realize that our long division yields:
\[
11x^3 - 13x
\]

In conclusion, the result of the polynomial division of \( 33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x \) by \( 3x^2 + 2x - 1 \) is:
\[
\boxed{11x^3 - 13x}
\]

There is no remainder in this case.