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Chapter 2: Problem 42

Use synthetic division to divide. \(\frac{-3 x^{4}}{x+2}\)

Short Answer

Expert verified
The quotient when \(-3x^{4}\) is divided by \(x+2\) using synthetic division is \(-3x^{3} + 6x^{2} - 12x + 24\).

Step by step solution

01

Set up the Synthetic Division

First, set up the synthetic division, where the divisor is \(x+2\) and dividend is \(-3x^{4}\). The root of the divisor \(x+2=0\) is \(-2\). Write \(-2\) at the left, outside the synthetic division bar. Then inside the bar, from left to right, write the coefficients of the polynomial in descending order. So, for \(-3x^{4}\), the coefficients are [-3, 0, 0, 0, 0].
02

Perform Synthetic Division

Bring down the first coefficient (-3) and write below the line. Then multiply with root (-2) to get 6. Write this result under the second coefficient (0), then add them to get 6. Repeat the process for the remaining coefficients; multiply 6 by root (-2), write under the third coefficient (0), add them to get -12. Continue this process until you perform operations for all coefficients. The final array is [-3, 6, -12, 24, -48].
03

Interpret the Results

The final array corresponds to the coefficients of the terms in the quotient polynomial. Starting from the highest exponent which is the original exponent minus 1 (in this case, 4-1=3), the quotient polynomial is \(-3x^{3} + 6x^{2} - 12x + 24\). The last value in the array, -48, is the remainder. But since the remainder is 0, our final quotient polynomial remains the same.

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