Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use synthetic division to divide. $$\left(2 x^{3}+14 x^{2}-20 x+7\right) \div(x+6)$$

Step-by-Step Solution

Answer

The result of the division \((2 x^{3}+14 x^{2}-20 x+7) \div(x+6)\) using synthetic division is \(2 x^{2} - 2 x - 8 - \frac{41}{x+6}\)

Step 1: Arrange the Coefficients for Synthetic Division

First, write down the coefficients of the dividend, which are the coefficients of the terms in order from highest to lowest degree. Here, the coefficients are \(2, 14, -20, 7\). Also, write down the number \(-6\), which is the opposite of the value of \(x\) from the divisor \(x + 6\).

Step 2: Perform the Synthetic Division Operation

Bring down the leading coefficient (2 in this case) to the bottom row. Multiply this number by \(-6\) and write the product in the next column. Then add the numbers in this column and write the result on the bottom row. Repeat this procedure until all the columns are filled.

Step 3: Interpret the Result

The last number in the bottom row is the remainder. The remaining numbers on the bottom row are the coefficients of the quotient. The degree of the terms in this quotient are one degree less than the degree of corresponding terms in the original dividend polynomial.

Step 4: Write the Result

The result of the division can be written in the form quotient + remainder/divisor. In this case, the quotient is \(2 x^{2} - 2 x - 8\) and the remainder is \(-41\), thus the result can be expressed as \(2 x^{2} - 2 x - 8 - \frac{41}{x+6}\)