Find the partial fraction decomposition of each rational expression. (2 x+4)/(x^3-1)

Name: Problem 22, Chapter 8: College Algebra
Uploaded: 2020-06-20T20:39:58-08:00
Duration: 6 min 2 s
Channel: Jason Schachner
Description: Find the partial fraction decomposition of each rational expression. (2 x+4)/(x^3-1)

Find the partial fraction decomposition of each rational...

Key Concepts

Setting Up and Solving Systems of Equations

After expressing the rational expression as a sum of fractions with undetermined coefficients, one equates the numerators on both sides by eliminating the common denominator. This process creates a system of linear equations for the coefficients, which, once solved, completes the decomposition. This systematic approach is fundamental to ensuring accuracy in the partial fractions method.

Decomposition Involving Linear and Quadratic Factors

When decomposing into partial fractions, the factors in the denominator determine the form of each term in the decomposition. Linear factors result in simple constants in the numerator, whereas irreducible quadratic factors typically require a linear expression in the numerator. This ensures that the sum of the parts correctly reconstructs the original expression.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational expression into a sum of simpler fractions. This technique is particularly useful for integration, solving differential equations, and simplifying algebraic expressions, as it allows one to work with more manageable terms that have simpler denominators.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of simpler, irreducible polynomials. This step is crucial in partial fraction decomposition because it simplifies the denominator into factors over which the numerator can be distributed in separate fractions. Common factorization techniques include the difference of cubes, difference of squares, and quadratic factorization.

Transcript

00:01 In order to attain our partial fraction decomposition of the rational expression, we must first look at the denominator.

00:07 Now, x -cube minus 1 is factorable.

00:12 You can factor an x -minus 1 out, and you're left with x squared plus x -plus 1, all under 2x plus 4.

00:27 So that's going to equal sum number a.

00:31 Now we have a repeated and we have two non -repeated functions.

00:36 1 is irreducible, right? so x minus 1 is our first function that's not repeated.

00:41 So we can just have a over x minus 1.

00:43 Now, since x squared plus x plus 1 is a non -repeating irreducible function, we can set that with some number b times x plus some number c, all over our non -repeating irreducible function, x squared plus x plus 1.

01:01 Now let's multiply both sides of the equal sign by our denominator of x minus 1 times x squared plus x plus 1 to get 2x plus 4 is equal to a times x squared plus x plus 1 plus plus bx plus c times x minus 1.

01:28 Now let's multiply everything out on the right side.

01:32 So we're going to get 2x plus 4 which is equal to a x squared plus a x plus a plus a plus b when we multiply these out we get bx squared minus bx plus cx minus c now let's group our like terms on the right side so let's rewrite this as okay so 2x plus 4 is equal to so let's factor an x squared out of here so is equal to a plus a plus plus b times x squared plus now let's factor our x's looks like we have three of them so plus a minus b plus c times x plus what's left an a and a minus c so plus a minus c now let's equate our coefficients on the right side and the left of our equal sign so there is no x squared on the left side, so that results in our a plus b coefficient on the right side to equal zero.

02:55 Our a minus b plus c is going to equal the coefficient of x on the left side, so that'll equal two.

03:02 So a minus b plus c equals two.

03:08 And finally, our a minus c coefficient will correspond with four, since there's no x is in either side...

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