Problem 19 - College Algebra

Question

Divide using synthetic division. $$\left(3 x^{2}+7 x-20\right) \div(x+5)$$

Step-by-Step Solution

Answer

The result of the division \((3 x^{2}+7 x-20) \div(x+5)\) using synthetic division is \(3x - 8 + \frac{20}{x+5}\)

Step 1: Setup for Synthetic Division

To set up for synthetic division, write the coefficients of the dividing polynomial (here, \(3 x^{2}+7 x-20\) ) in order and arrange them horizontally. To do this, write down the coefficients of each term in descending order of power. Thus, the coefficients would be 3, 7, and -20. Below and to the left of them, write the number which makes the divisor zero. In this case, it is -5 since that is what renders \(x+5\) equal to zero.

Step 2: Perform the Synthetic Division

Begin by bringing down the leading coefficient (which is 3 in this case) to the bottom row. Afterwards, multiply this number by -5 (the number you wrote to the left), and write the result (i.e., -15) under the second coefficient (i.e., 7) in the top row, then add straight down to get -8. Repeat this process: Multiply -8 by -5 to get 40 and write this under -20, then add to get 20.

Step 3: Interpret the Result

The numbers in the bottom row represent the coefficients of the polynomial result of the division. The final result chronicles from the highest power (one less than the power of the original polynomial) to the lowest. Consequently, the result of the synthetic division is \(3x - 8 + \frac{20}{x+5}\). Please note that the final term, 20, is the remainder and is divided by the original divisor, \(x+5\).