Divide using synthetic division : (x7 - 128)/(x - 2)

Given : Division: (x7 - 128)/(x - 2).

Answered: Divide using synthetic division : (x7 -…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter4: Polynomial And Rational Functions

Section4.3: The Remainder And Factor Theorems; Synthetic Division

Problem 108E

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Divide using synthetic division : (x⁷ - 128)/(x - 2)

Expert Solution

Step 1

Given :

Division: (x⁷ - 128)/(x - 2).

Step 2

By using synthetic division divide $x^{7} - 128$ by $x - 2$ ,

Step :1 Write down the first coefficient without changes.

$\begin{matrix} 2 \end{matrix} \begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & - 128 \end{matrix} 1$

step 2: Multiply the entry in the left part of the table by the last entry in the result row. Add the obtained result to the next coefficient of the dividend, and write down the sum and repeat it for all.

$\begin{matrix} 2 \end{matrix} \begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & - 128 \\ 2 & 4 & 8 & 16 & 32 & 64 & 128 \end{matrix} \begin{matrix} 1 & 2 & 4 & 8 & 16 & 32 & 64 & 0 \end{matrix}$

We have completed the table and have obtained the following resulting coefficients: $1, 2, 4, 8, 16, 32, 64, 0$ $1, 2, 4, 8, 16, 32, 64, 0$ .

All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.

Therefore, the quotient is $x^{6} + 2 x^{5} + 4 x^{4} + 8 x^{3} + 16 x^{2} + 32 x + 64$ and the remainder is 0.

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