Question
long division 33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x by 3x^2 + 2x - 1 in standard form
Answers
GPT-4o mini
To perform long division of the polynomial \(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x\) by the polynomial \(3x^2 + 2x - 1\), we follow the long division process similar to numerical long division. Here's how we do it step by step:
1. **Set up the division**:
\[
\text{Dividend: } 33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x
\]
\[
\text{Divisor: } 3x^2 + 2x - 1
\]
2. **Divide the leading term** of the dividend by the leading term of the divisor:
\[
\frac{33x^5}{3x^2} = 11x^3
\]
3. **Multiply the entire divisor** by \(11x^3\) and subtract from the dividend:
\[
(11x^3)(3x^2 + 2x - 1) = 33x^5 + 22x^4 - 11x^3
\]
Now subtract this product from the original polynomial:
\[
(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x) - (33x^5 + 22x^4 - 11x^3) = -50x^3 + 11x^3 - 26x^2 + 13x = -39x^3 - 26x^2 + 13x
\]
4. **Repeat the process** with the new dividend \(-39x^3 - 26x^2 + 13x\):
Divide the leading term:
\[
\frac{-39x^3}{3x^2} = -13x
\]
Multiply the divisor by \(-13x\):
\[
(-13x)(3x^2 + 2x - 1) = -39x^3 - 26x^2 + 13x
\]
Subtract:
\[
(-39x^3 - 26x^2 + 13x) - (-39x^3 - 26x^2 + 13x) = 0
\]
5. Now we see that the remainder is \(0\). We can stop here since there are no more terms to divide.
Putting everything together, the result of the long division can be summarized as:
\[
\text{Quotient: } 11x^3 - 13x
\]
Thus, the final answer in standard form is:
\[
\frac{33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x}{3x^2 + 2x - 1} = 11x^3 - 13x
\]
1. **Set up the division**:
\[
\text{Dividend: } 33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x
\]
\[
\text{Divisor: } 3x^2 + 2x - 1
\]
2. **Divide the leading term** of the dividend by the leading term of the divisor:
\[
\frac{33x^5}{3x^2} = 11x^3
\]
3. **Multiply the entire divisor** by \(11x^3\) and subtract from the dividend:
\[
(11x^3)(3x^2 + 2x - 1) = 33x^5 + 22x^4 - 11x^3
\]
Now subtract this product from the original polynomial:
\[
(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x) - (33x^5 + 22x^4 - 11x^3) = -50x^3 + 11x^3 - 26x^2 + 13x = -39x^3 - 26x^2 + 13x
\]
4. **Repeat the process** with the new dividend \(-39x^3 - 26x^2 + 13x\):
Divide the leading term:
\[
\frac{-39x^3}{3x^2} = -13x
\]
Multiply the divisor by \(-13x\):
\[
(-13x)(3x^2 + 2x - 1) = -39x^3 - 26x^2 + 13x
\]
Subtract:
\[
(-39x^3 - 26x^2 + 13x) - (-39x^3 - 26x^2 + 13x) = 0
\]
5. Now we see that the remainder is \(0\). We can stop here since there are no more terms to divide.
Putting everything together, the result of the long division can be summarized as:
\[
\text{Quotient: } 11x^3 - 13x
\]
Thus, the final answer in standard form is:
\[
\frac{33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x}{3x^2 + 2x - 1} = 11x^3 - 13x
\]