To divide the polynomial [tex]\(33 x^5 + 22 x^4 - 50 x^3 - 26 x^2 + 13 x \)[/tex] by [tex]\(3 x^2 + 2 x - 1\)[/tex] using long division, follow these steps:
Step 1: Set up the division:
Arrange the polynomials in standard form, ensuring they are written in descending order of their exponents. The numerator (dividend) is [tex]\(33 x^5 + 22 x^4 - 50 x^3 - 26 x^2 + 13 x\)[/tex] and the denominator (divisor) is [tex]\(3 x^2 + 2 x - 1\)[/tex].
Step 2: Divide the first term:
Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[
\frac{33 x^5}{3 x^2} = 11 x^3
\][/tex]
This gives the first term of the quotient.
Step 3: Multiply and subtract:
Multiply [tex]\(11 x^3\)[/tex] by the entire divisor [tex]\(3 x^2 + 2 x - 1\)[/tex] and subtract the result from the original numerator:
[tex]\[
11 x^3 \cdot (3 x^2 + 2 x - 1) = 33 x^5 + 22 x^4 - 11 x^3
\][/tex]
Subtract this product from the original numerator:
[tex]\[
(33 x^5 + 22 x^4 - 50 x^3 - 26 x^2 + 13 x) - (33 x^5 + 22 x^4 - 11 x^3)
\][/tex]
This simplifies to:
[tex]\[
-50 x^3 - 26 x^2 + 13 x - (-11 x^3) = -50 x^3 + 11 x^3 - 26 x^2 + 13 x = -39 x^3 - 26 x^2 + 13 x
\][/tex]
Step 4: Repeat the process:
Now we repeat the process with [tex]\(-39 x^3 - 26 x^2 + 13 x\)[/tex]:
[tex]\[
\frac{-39 x^3}{3 x^2} = -13 x
\][/tex]
Multiply [tex]\(-13 x\)[/tex] by the entire divisor and subtract:
[tex]\[
-13 x \cdot (3 x^2 + 2 x - 1) = -39 x^3 - 26 x^2 + 13 x
\][/tex]
Subtract this product from the new polynomial:
[tex]\[
(-39 x^3 - 26 x^2 + 13 x) - (-39 x^3 - 26 x^2 + 13 x) = 0
\][/tex]
Step 5: Obtain the quotient and remainder:
At this point, our remainder is [tex]\(0\)[/tex], and our quotient is the sum of terms we accumulated during the division steps: [tex]\(11 x^3 - 13 x\)[/tex].
Result:
Thus, the quotient is:
[tex]\[
Q(x) = 11 x^3 - 13 x
\][/tex]
And the remainder is:
[tex]\[
R(x) = 0
\][/tex]
So, the answer in standard form is:
[tex]\[
\boxed{11 x^3 - 13 x}
\][/tex]
