Question

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

Answer 1

To perform the long division of the polynomial \(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x\) by \(3x^2 + 2x - 1\), we will follow the long division steps for polynomials.

### Step 1: Set Up the Division

We write the dividend \(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x\) and the divisor \(3x^2 + 2x - 1\).

### Step 2: Divide the Leading Terms

1. **Divide the leading term of the dividend by the leading term of the divisor**:

\[
\frac{33x^5}{3x^2} = 11x^3
\]

2. **Multiply the entire divisor by \(11x^3\)**:

\[
11x^3(3x^2 + 2x - 1) = 33x^5 + 22x^4 - 11x^3
\]

3. **Subtract this from the original polynomial**:

\[
(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x) - (33x^5 + 22x^4 - 11x^3) = -50x^3 + 11x^3 - 26x^2 + 13x
\]

This simplifies to:

\[
-39x^3 - 26x^2 + 13x
\]

### Step 3: Repeat the Process

1. **Divide the leading term of the new polynomial by the leading term of the divisor**:

\[
\frac{-39x^3}{3x^2} = -13x
\]

2. **Multiply the entire divisor by \(-13x\)**:

\[
-13x(3x^2 + 2x - 1) = -39x^3 - 26x^2 + 13x
\]

3. **Subtract from the new polynomial**:

\[
(-39x^3 - 26x^2 + 13x) - (-39x^3 - 26x^2 + 13x) = 0
\]

### Conclusion of Long Division

After the subtraction, we find that there are no remaining terms. Thus, all terms have canceled out.

### Result

The quotient when dividing \(33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x\) by \(3x^2 + 2x - 1\) is:

\[
**11x^3 - 13x**
\]

The final answer is:

\[
\boxed{11x^3 - 13x}
\]