Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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 1: 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).

Step 2: 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.

Step 3: 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.

Step 4: Repeat the Division

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

Step 5: 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.