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:
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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.
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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.
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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: \[ (33x^5 + 22x^4 - 50x^3) - (33x^5 + 22x^4 - 11x^3) = (-50x^3 + 11x^3) - 26x^2 + 13x = -39x^3 - 26x^2 + 13x \]
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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.
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Multiply and Subtract Again: \[ -13x(3x^2 + 2x - 1) = -39x^3 - 26x^2 + 13x \] Subtract this from the current polynomial: \[ (-39x^3 - 26x^2 + 13x) - (-39x^3 - 26x^2 + 13x) = 0 \]
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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.