Problem 44 - Algebra and Trigonometry

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 solution to the long division of \(x^{5}\) by \(x^{2}-4 x+4\) yields a quotient and remainder. The remainder is further decomposed into simpler fractions using the method of partial fraction decomposition.

Step 1: Long Division of Polynomials

First, divide \(x^{5}\) by \(x^{2}-4x+4\) using polynomial long division. Matching the highest degree terms, write the quotient that when multiplied back by \(x^{2}-4x+4\) will yield the first term of the dividend. Then subtract the result of the multiplication from the original polynomial to obtain the new polynomial for the next round of division.

Step 2: Calculation of Remainder

Continue the long division process until you can't divide any more, i.e., the degree of the remainder polynomial is less than that of the divisor. The polynomial thus obtained will be the remainder.

Step 3: Formulation of Partial Fraction Decomposition

The decomposition of the remainder term involves expressing it as a sum of simpler fractions.

Step 4: Solving for Unknowns

Having divided out the fraction into a main polynomial part and a remainder expressed as a fraction, equate this to the original expression and solve for the unknown constants. This will involve equating coefficients and solving simultaneous equations.

Key Concepts

Polynomial Long Division

Imagine you have a big sheet of paper and a pair of scissors. You want to cut the paper into smaller pieces of a specified size. This is quite similar to polynomial long division. The goal is to divide a larger polynomial by another smaller polynomial, to see how many times the smaller polynomial fits into the larger one. Start by taking the first term of the dividend (the polynomial you are dividing) and the first term of the divisor (the polynomial you are dividing by). Use only the leading coefficients — the numbers in front of the variables with the highest power.

• Calculate how many times the first term of the divisor fits into the first term of the dividend, and write that as part of the quotient.
• Multiply the entire divisor by this quotient term and subtract the result from the original polynomial.
• This new polynomial becomes your "new" dividend.
• Repeat the process with the new dividend.

Eventually, you’ll end up with a remainder that’s smaller in degree than the divisor. That’s when you stop dividing.

Remainder Theorem

The Remainder Theorem is like a magical guide when dealing with polynomials. It helps you understand remainders without having to perform the entire long division. Sounds cool, right?When you divide a polynomial by a binomial of the form \((x-a)\), the remainder of the division is simply the value of the polynomial when \(x = a\). This theorem can immensely speed up your polynomial division tasks.Here’s how it works in simpler terms:

• Take your polynomial and substitute \(x = a\) into it.
• The result gives you the remainder of the division.

In our exercise, the polynomial division continues until the degree of the remainder is less than the degree of the divisor. Understanding this can save you both time and effort.

Rational Expressions

Rational expressions are like the teenagers of the math world—expressions with a numerator and denominator that can create beautiful complexities. They’re essentially fractions where the top and bottom are polynomials. The goal is often to simplify these expressions by factoring and canceling similar terms in the numerator and denominator. Partial fraction decomposition comes into play with rational expressions when you want to break down a complex fraction into simpler, more manageable pieces. This approach is important because it helps to:

• Simplify complex rational expressions.
• Make calculus operations such as integration more manageable.

In our exercise, after finding the remainder from the division, the task brings you to express it as simpler fractions, which is why understanding rational expressions and their decompositions is so helpful.