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Chapter 3: Problem 31

Divide using synthetic division. $$\frac{2 x^{5}-3 x^{4}+x^{3}-x^{2}+2 x-1}{x+2}$$

Short Answer

Expert verified
The result of the division \(\frac{2 x^{5}-3 x^{4}+x^{3}-x^{2}+2 x-1}{x+2}\) is \(2x^{4}-7x^{3}+15x^{2}-30x+60; remainder -3\)

Step by step solution

01

Set Up The Division

Set up the division by writing the coefficients of the polynomial equation to be divided in order, from highest to lowest exponent of x. The polynomial \(2x^{5}-3x^{4}+x^{3}-x^{2}+2x-1\) has coefficients 2, -3, 1, -1, 2, -1. The divisor \(x+2\) will be represented by the value of x which makes the divisor equals to zero, that is x = -2.
02

Carry Out the Division

Perform the synthetic division as follows: Write down the first coefficient (2, in this case). Multiply it by the value representing the divisor (-2), and write this product directly underneath the next coefficient (-3). Add the numbers in this new column, -3 plus -4, to get -7. Repeat the process: Multiply -7 by -2 to get 14, write this under the 1, and add to get 15. Continue this pattern until you have processed all the coefficients of the original polynomial.
03

Write the Result

After performing the synthetic division, the result is given by reading off the numbers on the bottom as the new coefficients. Since we started with a fifth-degree polynomial, the quotient will be a fourth-degree polynomial. The final coefficient -3 is the remainder. Therefore the result is \(2x^{4}-7x^{3}+15x^{2}-30x+60; remainder -3\)

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