91Ó°ÊÓ

Setup the synthetic division by writing the coefficients of the dividend (the polynomial to be divided) to the right of a vertical line. Write the coefficients in order of descending power. The dividend is \(x^{7}-128\) which should be written as \(x^{7} + 0x^{6} + 0x^{5} + 0x^{4} + 0x^{3}+ 0x^{2}+ 0x^{1}-128\), therefore the coefficients are 1, 0, 0, 0, 0, 0, 0, -128. After this, put the root of the divisor (the value opposite of -a from the divisor \(x-a\)) to the left of the vertical line. For \(x-2\), the root is 2. Write the root and the coefficients in this form: \[2 | 1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0\quad -128\]

Drop down the leading coefficient to begin the synthetic division. This becomes the leading coefficient of the quotient.\[2 | 1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0\quad -128\]\[ \quad \downarrow \]

Multiply the root (2) by the value just dropped down (1). Write the product underneath the second number in the top row (0) and add the column, writing the total beneath. Repeat this operation for each number in the top row.\[2 | 1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0\quad -128\]\[ \quad \quad \quad 2 \quad \quad 4 \quad \quad 8 \quad \quad 16 \quad \quad 32 \quad \quad 64 \quad \quad 128 \]\[ \quad \quad \quad 2 \quad \quad 4 \quad \quad 8 \quad \quad 16 \quad \quad 32 \quad \quad 64 \quad \quad \quad 0 \]In this case, the remainder is 0 because the original polynomial is perfectly divisible by \(x-2\). The numbers in the bottom row represent the coefficients of the quotient, starting from one degree less than the dividend started.

The quotient is presented by associating the coefficients from the bottom row with the appropriately decreasing powers of \(x\), starting from \(x^6\) (one degree less than the dividend). This gives the result: \(x^{6} + 2x^{5} + 4x^{4} + 8x^{3} + 16x^{2} + 32x +64\)