Problem 46 - College Algebra

Question

Perform each long division and write the partial fraction decomposition of the remainder term. $$\frac{x^{4}+2 x^{3}-4 x^{2}+x-3}{x^{2}-x-2}$$

Step-by-Step Solution

Answer

The expression \(\frac{x^{4}+2 x^{3}-4 x^{2}+x-3}{x^{2}-x-2}\) after the process of polynomial long division and partial fraction decompositions becomes \(x^2 + 3x + 2 - \frac{5}{x-2} - \frac{1}{x + 1}\).

Step 1: Polynomial Long Division

We perform long division on \( \frac{x^{4}+2 x^{3}-4 x^{2}+x-3}{x^{2}-x-2} \). This gives us the quotient \(x^2 + 3x + 2\) and remainder \(-6x + 1\). The fraction therefore becomes \(x^2 + 3x + 2 + \frac{-6x + 1}{x^{2}-x-2}\).

Step 2: Factorize the Denominator

Factorize the denominator \(x^{2}-x-2\). This turns to \((x - 2)(x + 1)\). In this step, we identified what the factors of the quadratic equation are, these will be used in the next step that specific to performing partial fraction decomposition.

Step 3: Partial Fraction Decomposition

We rewrite \(\frac{-6x + 1}{x^{2}-x-2}\) as \(\frac{A}{x - 2} + \frac{B}{x + 1}\), where A and B are constants to be found. Multiplying through by \(x^{2}-x-2\) gives the equation \(-6x + 1 = A(x + 1) + B(x - 2)\). Solving this system, we find \(A = -5\) and \(B = -1\). The original fraction can therefore be rewritten as \(x^2 + 3x + 2 - \frac{5}{x-2} - \frac{1}{x + 1}\)

Key Concepts

Polynomial Long Division

Polynomial long division is akin to the long division process you learned with numbers, but it's used for dividing a polynomial by another polynomial. It's an essential technique in algebra, especially when dealing with higher-degree polynomials. The goal here is to break down a complex fraction into a simpler polynomial plus a remainder, much like an integer division.

To perform this division, we start by dividing the first term of the numerator by the first term of the denominator and multiply the entire denominator by that result. We then subtract this from the original polynomial and bring down the next term. We repeat this process until the degree of the remainder is less than the degree of the denominator.

This process can be helpful in simplifying algebraic expressions and solving equations, making it an indispensable tool in college algebra. It's crucial to master this technique for practical use in algebraic manipulation and calculus.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. Factors are polynomials of lower degrees that multiply together to give the original polynomial. This is a vital skill in algebra because it simplifies many algebraic processes from solving quadratic equations to simplifying algebraic fractions and partial fraction decomposition.

There are various methods of factoring, such as pulling out a greatest common factor, using the difference of squares, and factoring trinomials, among others. Effective factoring often requires looking for patterns and using specific formulas or methods suited to the particular form of the polynomial. This can be one of the trickier skills for students to master but is incredibly rewarding and widely applicable in more advanced mathematics.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or denominator contain algebraic expressions. Similar to numerical fractions, they can be added, subtracted, multiplied, divided, and simplified. Simplifying complex algebraic fractions often involves factoring polynomials and polynomial long division.

Manipulating algebraic fractions is a cornerstone of algebra and pre-calculus. One common goal in working with algebraic fractions is to express them in simplest form, where the numerator and denominator share no common factors other than 1. Learning to work comfortably with algebraic fractions opens the door to solving equations that involve ratios of polynomials and understanding the behavior of rational functions.

College Algebra

College algebra is a course that typically follows the curriculum of high school algebra II, delving deeper into algebraic concepts and their applications. It lays a strong foundation for all future courses in mathematics, science, and engineering. Key topics often include polynomial operations, factoring, algebraic fractions, functions, and their graphs, systems of equations, and sometimes the basics of analytic geometry.

College algebra aims to equip students with robust problem-solving skills that will be useful across various disciplines. It's not merely about learning procedures but understanding and applying the underlying principles of algebra to diverse and complex problems. Topics like polynomial long division and partial fractions are staples of this subject and serve as building blocks for more advanced topics in calculus and beyond.