Find the partial fraction decomposition of (2)/(x^2-a^2). |...

Name: Problem 46, Chapter 7: Precalculus: Graphical, Numerical, Algebraic
Uploaded: 2019-07-13T22:44:28-08:00
Duration: 3 min 56 s
Channel: Weston Bell-Geddes
Description: Find the partial fraction decomposition of (2)/(x^2-a^2).

Key Concepts

Difference of Squares Factorization

This is a specific factoring technique used when a polynomial can be expressed as the difference between two perfect squares. Recognizing and applying this pattern simplifies the denominator into a product of two binomials, which is a common requirement for setting up partial fractions.

Partial Fraction Decomposition

This is an algebraic technique used to rewrite a rational function—one polynomial divided by another—as the sum of simpler rational expressions, usually with linear or irreducible quadratic denominators. This method is especially useful for integration and solving differential equations where handling the simpler fractions is more straightforward.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. Decomposing these functions into partial fractions can simplify operations such as integration, differentiation, and the solving of equations, as working with simpler expressions is often much more manageable.

Factoring Polynomials

In the context of partial fraction decomposition, factoring the denominator of the rational function is a critical step. It involves expressing the denominator as a product of simpler polynomials, which then allows the original function to be split into a sum of fractions where each term corresponds to one of these factors.

Transcript

00:01 We're asked to find the partial fraction decomposition of 2 all over x squared minus a squared.

00:07 So the first thing i'm going to do is factor denominator here to 2 over x minus a times x or a plus x.

00:19 The next thing i'm going to do is write this in the partial fraction decomposition form, which is a over the first item, which is x minus a plus b over the second.

00:32 Item, which is a plus x.

00:37 Perfect.

00:38 So now i'm going to want to combine this fraction, these two fractions over here.

00:42 So in combining these two fractions, i'm going to do cross multiplication.

00:47 I'm going to get something looks like this.

00:48 A times a plus x all over x minus a times a plus x plus b times x minus a all over a plus x, parentheses x minus a, beautiful so now that these two sides are i are the same here and here we can go ahead and add them across and then cancel out the bottoms here so doing that looks similar like this 2 equals a a plus x plus b x minus a next i'm gonna expand this to a times a lowercase a that is be cautious with that, times capital a times x, plus capital b times x minus b times lowercase a.

01:49 This is all equal to two still.

01:51 Now i'm going to collect my terms.

01:53 So, this equals 2 equals a plus b times x, plus a times a minus b times a.

02:03 So now with that we have this, we can write our system of equations...

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