Question: Use long division to divide the polynomial $33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x$ by $3x^2 + 2x - 1$. Write your answer in standard form. (1 point)
1. **State the problem:** Divide the polynomial $$33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x$$ by $$3x^2 + 2x - 1$$ using long division.
2. **Recall the formula and rules:** Polynomial long division is similar to numerical long division. We divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient, then multiply the entire divisor by this term and subtract from the dividend. Repeat until the degree of the remainder is less than the divisor.
3. **Step 1:** Divide the leading term $$33x^5$$ by $$3x^2$$:
$$\frac{33x^5}{3x^2} = 11x^3$$
4. **Step 2:** Multiply the divisor by $$11x^3$$:
$$11x^3(3x^2 + 2x - 1) = 33x^5 + 22x^4 - 11x^3$$
5. **Step 3:** Subtract this from the original polynomial:
$$\left(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x\right) - \left(33x^5 + 22x^4 - 11x^3\right) = -39x^3 - 26x^2 + 13x$$
6. **Step 4:** Divide the new leading term $$-39x^3$$ by $$3x^2$$:
$$\frac{-39x^3}{3x^2} = -13x$$
7. **Step 5:** Multiply the divisor by $$-13x$$:
$$-13x(3x^2 + 2x - 1) = -39x^3 - 26x^2 + 13x$$
8. **Step 6:** Subtract this from the previous remainder:
$$\left(-39x^3 - 26x^2 + 13x\right) - \left(-39x^3 - 26x^2 + 13x\right) = 0$$
9. **Step 7:** Since the remainder is zero, the division is exact.
10. **Final answer:** The quotient is $$11x^3 - 13x$$.
**Answer in standard form:** $$11x^3 - 13x$$.