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Chapter 3: Problem 20

Divide using synthetic division. $$\left(5 x^{2}-12 x-8\right) \div(x+3)$$

Short Answer

Expert verified
The result of the division is \(5x - 27 - \frac{11}{x+3}\)

Step by step solution

01

Setup the Synthetic Division

Rewrite the divisor and the dividend in the synthetic division format. The coefficient of \(x\) in the divisor \(x+3\) is -3 (since it's \(x - (-3)\)). List the coefficients of the dividend \(5x^2-12x-8\) in a row: 5, -12, and -8.
02

Start the Synthetic Division

First, bring down the leading coefficient (5) to the bottom row. Multiply the number you just wrote (5) by -3 (the coefficient of \(x\) in the divisor), write the result under the next coefficient, then add down. Write the result in the bottom row.
03

Continue the Synthetic Division

Repeat the process from step 2: multiply the number you just wrote (-27) by -3, result under the next (and in this case last) coefficient, then add down, resulting in -11. This is the remainder.
04

Write Down the Final Result

The bottom row represents the coefficients of the result of the division. From left to right, the coefficients correspond to the terms of a polynomial of degree one less than the original polynomial. Our original is 2nd degree, so our result is 1st degree. This means our solution is \(5x-27\) with a remainder of -11, or expressed as \(5x - 27 - \frac{11}{x+3}\)

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