Problem 20 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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 \(5x - 27 - \frac{11}{x+3}\)