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Chapter 3: Problem 29

Divide using synthetic division. $$\frac{x^{4}-256}{x-4}$$

Short Answer

Expert verified
The result of the division \(\frac{x^{4}-256}{x-4}\) is \(x^{3}+4x^{2}+16x+64\).

Step by step solution

01

Set up the synthetic division

It is seen that \(x-4\) is the divisor which implies the zero for this factor is \(x = 4\). Write this value off to the side, set up a division and list the coefficients for each term in the polynomial. In this case, our polynomial is \(x^{4}-256\). The coefficients for our terms are \(1, 0, 0, 0, -256\). If there are any missing terms in the polynomial we sort it by filling the missing coefficients with zero.
02

Synthetic division process

Start the process by bringing down the leading coefficient of \(1\). Then, multiply it by the zero of \(4\), and write the result under the next coefficient. Next, add the numbers in that column, multiply by \(4\) again and continue this process until the last number is calculated. The numbers on the bottom row of our table would be \(1, 4, 16, 64, 256\). The final number is the remainder and in this case, it is \(0\), hence there is no remainder.
03

Write the result

Finally, we write our answer using the coefficients we have derived from our synthetic division. Always remember, we start one degree lower than the original dividend. Hence, our answer becomes \(x^{3}+4x^{2}+16x+64\).

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