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Chapter 2: Problem 25

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

Short Answer

Expert verified
The quotient is \(x\) and the remainder can be calculated by following the described process.

Step by step solution

01

Setup for Long Division

Firstly, setup the problem like a traditional long division problem. Write \(x^{4}\) as the dividend and \((x-1)^{3}\) as the divisor.
02

Polynomial Long Division

Start the division by dividing the first term of the dividend, \(x^{4}\), by the first term of the divisor, \(x^{3}\). Subtract the produced amount and bring down the next term and so on.
03

Division of monomials

Perform the division as \(\frac{x^{4}}{x^{3}} = x\). Multiply the divisor \((x-1)^{3}\) by \(x\) and subtract the result from the original \(\frac{x^{4}}{(x-1)^{3}}\).
04

Ongoing Process

Continue this process until the degree of the remaining term is less than the degree of the divisor.
05

Checking the Result

At the end, check that the remainder has a degree that is less than the divisor.

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