Find the partial fraction decomposition for each rational expression. = (1)/(x^2(x^2-2))

Name: Problem 20, Chapter 9: Precalculus Student Solutions Manual 5th
Uploaded: 2020-12-08T19:00:10-08:00
Duration: 3 min 58 s
Channel: Nick Johnson
Description: Find the partial fraction decomposition for each rational expression. = (1)/(x^2(x^2-2))

Find the partial fraction decomposition for each rational...

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding these expressions is key as many algebraic techniques, including partial fraction decomposition, rely on the manipulation and factorization of such expressions.

Polynomial Factorization

Factorization is the process of breaking down a polynomial into a product of simpler polynomials or irreducible factors. This is a crucial step in partial fraction decomposition, as it allows one to express the denominator in terms of factors that can be individually addressed during the decomposition process.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complicated rational expression as a sum of simpler fractions with denominators that are usually linear or quadratic. This technique is particularly useful in integration, solving differential equations, and simplifying algebraic expressions.

Repeated Factors

When a factor in the denominator appears more than once, it is known as a repeated factor. The decomposition must account for each repetition by including separate terms for each power of the repeated factor, ensuring a complete and correct partial fraction representation.

Equating Coefficients

This is a method used in partial fraction decomposition where after expressing the rational expression as a sum of simpler fractions, both sides of the equation are rewritten in a common denominator. The coefficients of corresponding powers of the variable are then equated, allowing one to solve for the unknown constants in the numerators.

Transcript

00:01 Okay, so here we have one over x squared times the quantity, x squared minus two, which we notice is a proper fraction here because we have the numerator has a lower degree than the denominator.

00:12 And, well, we're already kind of factored, but we can factor this further, right? x squared minus two factors as x plus root two times x minus root two.

00:21 So then we have, well, we have our first constant a over, we have an x squared.

00:26 We have to account for the square.

00:28 So we have a over x and then plus b over x squared and then plus c over our next factor, which would be x plus root 2, plus d over our next factors be x minus root 2.

00:40 Okay.

00:41 So now to go ahead and clear the denominator, we have to multiply both sides of our equation by x squared times x plus root 2 times x minus root 2.

00:55 So that gives us 1 is equal to a x times x squared minus 2 plus b times x squared minus 2 plus c times x squared 2 plus d times x squared times the quantity x plus root 2.

01:13 Okay, but if we just substitute in 0 for x, then we see that 1 is just equal to b times negative 2.

01:21 Okay, well, just to find negative 2 here, and we see that b, right, this implies that b is equal to, well, negative 1⁄2.

01:31 So that b is equal to negative 1⁄2.

01:34 Okay.

01:37 And then if we substitute in square root of 2 for x, we have that 1 is equal to c times 2 times negative 2 times the square root of 4...

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