Problem 44 - College Algebra

Question

Perform each long division and write the partial fraction decomposition of the remainder term. $$\frac{x^{5}}{x^{2}-4 x+4}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{x^{5}}{x^{2}-4 x+4}\) is \(x^{3}+4x^{2}+16x +64 + \frac{64}{x-2}+ \frac{64}{(x-2)^2}\).

Step 1: Polynomial Long Division

Perform long division of the polynomials \(x^{5}\) by the quadratic polynomial \(x^{2}-4 x+4\). Divide the highest degree term of numerator by the highest degree term of the denominator. Multiply the result by the denominator and subtract it from the numerator. Repeat this procedure until the degree of the remainder is less than the degree of divisor.

Step 2: Write the Result of Division

As a result of the long division of polynomials, we get \(x^{3}+4x^{2}+16x +64\) with a remainder of \(256.\) It can be written as \(x^{3}+4x^{2}+16x +64 + \frac{256}{x^{2}-4 x+4}\).

Step 3: Partial Fraction Decomposition

Perform the partial fraction decomposition of the remainder term \(\frac{256}{x^{2}-4 x+4}\). Set the denominator equal to zero \(x^{2}-4 x+4 = 0\) to find the roots and then decompose the fraction.

Step 4: Write the Final Solution

The partial fraction decomposition of \(\frac{256}{x^{2}-4 x+4}\) is \(\frac{64}{x-2}+ \frac{64}{(x-2)^2}\). Thus, the final solution of the given problem is \(x^{3}+4x^{2}+16x +64 + \frac{64}{x-2}+ \frac{64}{(x-2)^2}\).

Key Concepts

Polynomial Long Division

Polynomial long division is similar to the long division you learned in grade school, just with polynomials instead of numbers. It's a systematic method for dividing a polynomial by another polynomial of a lower degree. In the given exercise, we must divide the polynomial \(x^5\) by a quadratic polynomial \(x^2 - 4x + 4\).

Here’s how it works step-by-step:

• Start by dividing the highest degree term of the dividend \(x^5\) by the highest degree term of the divisor \(x^2\), which gives \(x^3\).
• Multiply the entire divisor by \(x^3\) and subtract this product from your original dividend.
• Repeat this process using the new polynomial obtained after subtraction, continuing until the degree of the remainder is less than the degree of the divisor \(x^2 - 4x + 4\).

Once the division is complete, you'll reach a stage where you can't divide further, as the remainder is of lower degree than the divisor.

Remainder Term

The remainder term is what is left over after you have completed the polynomial long division. For the example \(\frac{x^5}{x^2-4x+4}\), the division gives us a polynomial quotient and a remainder of 256.

The long division process generates a result that can be expressed in the form:

• Quotient: The result of the division without any remainder.
• Remainder Term: A smaller polynomial or a constant that represents the leftover part after full division.

In our example, the quotient is \(x^3 + 4x^2 + 16x + 64\) with a remainder of 256. It is important to express this remainder over the original divisor to reflect its contribution to the entire division process. Hence, the remaining term is expressed as \(\frac{256}{x^2 - 4x + 4}\).

This remainder term can often be further simplified or examined using partial fraction decomposition.

Roots of Polynomial

Finding the roots of a polynomial is essential for many algebraic procedures, including partial fraction decomposition, as it helps to simplify complex rational expressions.

In the problem at hand, the denominator \(x^2 - 4x + 4\) needs to be set to zero to find its roots:

• Solve: \(x^2 - 4x + 4 = 0\)
• This simplifies to \((x-2)^2 = 0\)
• Giving us a repeated root at \(x = 2\)

Once you have the roots, they serve as critical points that allow us to decompose the remainder term further into simpler fractions. These repeated roots mean the polynomial can be broken down into partial fractions that involve powers of the factor \((x - 2)\).

Roots tell us about the behavior of polynomials and how they can be broken down, which helps in rewriting expressions in a more manageable form.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable present in it and plays a crucial role in operations like polynomial division or decomposition.

In the division \(\frac{x^5}{x^2-4x+4}\), the degree of the numerator is 5 and the degree of the denominator is 2. The process of long division continues until the degree of the remainder is less than the degree of the divisor.

Understanding the degree helps us comprehend:

• Order of Operations: The polynomial of higher degree is the one being divided.
• Ending the Division: The division process stops when the remainder has a lower degree than the divisor.
• Arrangement and Simplification: Decomposing into partial fractions requires competent handling of polynomials of various degrees to facilitate simplifying expressions.

The degree gives important guidance on how deep or complex computations can get with a given polynomial, and understanding it allows for strategic simplifications.