Problem 30 - College Algebra

Question

Divide using synthetic division. $$\frac{x^{7}-128}{x-2}$$

Step-by-Step Solution

Answer

The solution to the given division problem is \(x^6+2x^5+4x^4+8x^3+16x^2+32x+64 - \frac{64}{x-2}\)

Step 1: Setup the Synthetic Division

In order to use synthetic division, the divisor needs to be of the form \(x - c\) where \(c\) is a constant. That is the case here as \(x - 2\). The constant is 2. Set up the synthetic division by writing 2 to the left and the coefficients of \(x^{7}-128\) to the right. However, make sure to include zeroes for any missing powers of \(x\). So we rewrite the dividend as \(x^{7} + 0x^{6} + 0x^{5} + 0x^{4} + 0x^{3} + 0x^{2} + 0x -128 \)

Step 2: Performing Synthetic Division

In synthetic division, first of all, bring down the first coefficient 1 (from \(x^7\)). Then multiply it with 2 (constant in \(x-2\)) and write the result under next coefficient of 0 from \(x^6\). Add these two numbers (0 obtained from multiplication, 0 from the polynomial) and write the result underneath. Repeat the process for all coefficients.

Step 3: Write Out The Answer

The numbers on the bottom provide the coefficients for the powers of \(x\) of the quotient. The quotient is one degree less than the dividend (the polynomial we started with). The final answer will be \(x^6+2x^5+4x^4+8x^3+16x^2+32x+64 - 64/x-2\). Notice, that the remainder is negative, so it is subtracted