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Chapter 3: Problem 26

Divide using synthetic division. $$\left(x^{2}-6 x-6 x^{3}+x^{4}\right) \div(6+x)$$

Short Answer

Expert verified
The synthetic division of \(\left(x^{4}-6x^{3}-6x^{2}+x\right) \div (x+6)\) results in \(x^{3}-12x^{2}+66x-396 + \frac{2376}{x+6}\)

Step by step solution

01

Prepare the Division

Reorder the divisor \(x + 6\) into the form \(x - a\) by rewriting it as \(x - (-6)\). Order the polynomial in terms of descending exponent and fill zero for any missing terms. The polynomial thus becomes, \(x^{4}-6x^{3}-6x^{2}+0x+0\). Write these coefficient values (1, -6, -6, 0, 0) in reverse - considering any factor for x as \(a\) in \(x - a\), so a = -6. Write -6 in a box or circle beneath the line.
02

Begin Synthetic Division

Drop the leading coefficient (from the polynomial) down. That is, write the first coefficient of the polynomial - which is 1 - beneath the line.
03

Multiply and Add

Multiply -6 (from the box) by the leading 1 you wrote below the line. Write this product (-6) beneath the second number on top line. Add these numbers together, getting -12, and write this result beneath the line. Repeat this step for the rest of the numbers on the top line.
04

Write Out the Result

The numbers obtained in the last row are the coefficients of the quotient polynomial. Starting from one exponent degree less than the dividend polynomial had (here, from \(x^3\)), write out the equation: \(x^{3}-12x^{2}+66x-396 + \frac{2376}{x+6}\). This equation is the result of the synthetic division. The final term \(\frac{2376}{x+6}\) is the remainder, which is obtained by placing the last above the divisor.

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