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Chapter 2: Problem 44

Use synthetic division to divide. \(\frac{5-3 x+2 x^{2}-x^{3}}{x+1}\)

Short Answer

Expert verified
The result of the synthetic division is \(-x^2+x+2\) with a remainder of 3.

Step by step solution

01

Arrange the Coefficients

Arrange the coefficients of the polynomial in the order of decreasing powers of x. This gives us \(-1, 2, -3, 5\). Additionally, take the opposite sign of the constant in the denominator, which gives us \(-1\).
02

Begin Synthetic Division

Start the synthetic division process. Write down the first coefficient, -1, in the bottom row. Then multiply this number by the number derived from the denominator, -1, and write the result under the second coefficient in the top row. Add these numbers together and write the result in the bottom row.
03

Continue Synthetic Division

Continue step 2 with each subsequent coefficient. Remember to alternate between writing the result in the bottom row and multiplying by the number derived from the denominator, then adding to the next coefficient in the top row. Repeat until there are no more coefficients in the top row.
04

Interpret the Result

The numbers in the bottom row represent the coefficients of the polynomial quotient. The last number is the remainder. So this gives us \(-x^2+x+2\) with a remainder of 3.

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