Problem 28 - College Algebra

Question

Divide using synthetic division. $$\frac{x^{7}+x^{5}-10 x^{3}+12}{x+2}$$

Step-by-Step Solution

Answer

The result of the division is \(x^{6} - 2x^{5} + 4x^{4} - 8x^{3} + 16x^{2} - 32x + 64 - 116 \cdot \frac{1}{x+2}\)

Step 1: Preparation

Prepare the synthetic division setting. Write down the coefficients of the polynomial in decreasing order, and the additive inverse of the constant term from the divisor on the left side. Because we have absent powers of \(x\) (there's no \(x^6\), \(x^4\), \(x^2\), and \(x\)), their coefficients are 0. So we write the coefficients as follows: 1 (for \(x^7\)), 0 (for the missing \(x^6\)), 1 (for \(x^5\)), 0 (for the absent \(x^4\)), -10 (for \(x^3\)), 0 (for absent \(x^2\)), 0 (for absent \(x\)), and 12 (the constant). And, we write -2 on the left (as opposite of 2 in the divisor \(x+2\))

Step 2: Synthetic Division

Start with bringing down the first coefficient (1 in this case) as it is. Multiply this number with the number on the left side and write the product below the next coefficient. Then, add these numbers and repeat the multiply-add steps until the last coefficient.

Step 3: Write Down the Result

The bottom row of numbers are the coefficients of the result, starting with one degree lower than the original polynomial. So, the answer becomes \(x^{6} - 2x^{5} + 4x^{4} - 8x^{3} + 16x^{2} - 32x + 64 - \frac{116}{x+2}\). The last term, -116, is the remainder (dividend = divisor * quotient + remainder).

Step 4: Simplify the Result (if necessary)

Since in this case there is a negative number in the remainder, the task here is to rewrite \(-\frac{116}{x+2}\) as \(-116 \cdot \frac{1}{x+2}\). So the final result is \(x^{6} - 2x^{5} + 4x^{4} - 8x^{3} + 16x^{2} - 32x + 64 - 116 \cdot \frac{1}{x+2}\).