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Chapter 3: Problem 22

Use long division to divide. $$\frac{x^{4}}{(x-1)^{3}}$$

Short Answer

Expert verified
Dividing \(x^{4}\) by \((x-1)^{3}\) using synthetic division results in the expression \(x + 4x^2 - 12x +12 + \frac{-12}{(x-1)^3}\).

Step by step solution

01

Set up the division

Set up the division similar to the long division in arithmetic with \(x^{4}\) on the inside (dividend) and \((x-1)^{3} = x^3 - 3x^2 + 3x -1\) on the outside (divisor).
02

Begin Division

To begin the division, divide the first term of the dividend, \(x^{4}\), by the first term of the divisor, \(x^3\). Put the result, \(x\), above the line (this starts the quotient). Next, multiply \(x\) by each term in \((x-1)^{3}\) and subtract the result from the appropriate terms in the dividend. Write the result below the line.
03

Continue Division

After subtraction, bring down the next term from the dividend and repeat the same process by dividing this new dividend by the first term of the divisor.
04

Repeat the Division

Repeat this process until the degree of the remaining polynomial is less than that of the divisor.
05

Write the result in the appropriate form

The final result (quotient) will be a polynomial plus a fraction where the numerator's degree is less than the denominator's.

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