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Chapter 3: Problem 28

Divide using synthetic division. $$\frac{x^{7}+x^{5}-10 x^{3}+12}{x+2}$$

Short Answer

Expert verified
The result of the division is \(x^{6}-2x^{5}+4x^{4}-8x^{3}+16x^{2}-32x+64-\frac{116}{x+2}\).

Step by step solution

01

Write the Coefficients

Write the coefficients of the divisor and the polynomial to be divided in a horizontal line. Remember to include zeros for any missing terms. For the given problem, the divisor is \(x + 2\) and the polynomial is \(x^{7}+x^{5}-10 x^{3}+12\), so the coefficients would be [-2, 1, 0, 1, 0, -10, 0, 12]. The first number is negative since in the divisor, \(x + 2\), 2 is subtracted from x, making it negative.
02

Synthetic division process

Begin synthetic division: start by bringing down the first number (1). Then, multiply this number by -2 (the divisor), and write this result under the next coefficient. Add these two together and write the sum below. Repeat this process for the remaining coefficients.
03

Interpret the result

The numbers on the bottom row are the coefficients of the quotient polynomial. Typically, the first number is the coefficient for the \(x^{6}\) term, the second number is for the \(x^{5}\) term, and so on, all the way to the constant term, which is the last number. In this case, the result would be: \(x^{6}-2x^{5}+4x^{4}-8x^{3}+16x^{2}-32x+64-\frac{116}{x+2}\).

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