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Chapter 2: Problem 38

Use synthetic division to divide. \(\frac{x^{5}-13 x^{4}-120 x+80}{x+3}\)

Short Answer

Expert verified
The result of the synthetic division of \(x^{5}-13 x^{4}-120 x+80\) by \(x+3\) is \(x^{4} -16x^{3}+48x^{2}-144x+432 \, remainder \, -1196\)

Step by step solution

01

Setup the Synthetic Division

The first step in synthetic division is to set up the synthetic division tableau. Write down the coefficients of the dividend \(x^{5}-13 x^{4}-120 x+80\) which are [1, -13, 0, 0, -120, 80], note that zero coefficients are included for the missing powers of x in the polynomial. Next, write down -3 (which is the root of the divisor \(x+3\)) on the left outside the synthetic division tableau.
02

Carry out the Synthetic Division Operation

To proceed with the division, start by dropping down the leading coefficient to the bottom row. Multiply this number in the bottom row by -3 and write the result in the next column in the second row. Add the numbers in this column and repeat the process for all columns in the tableau.
03

Interpret the Synthetic Division Results

The bottom row of the tableau represents the coefficients of the quotient and remainder. The quotient is the polynomial with these coefficients of decreasing powers of x, starting with \(x^{4}\). The last element of the bottom row is the constant term of the remainder. In this case, the quotient is \(x^{4} -16x^{3}+48x^{2}-144x+432\) and the remainder is -1196.

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