Find the partial fraction decomposition. (2 x^3+2 x^2+4 x-3)/(x^4+x^2)

Name: Problem 20, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2020-11-13T19:02:43-08:00
Duration: 4 min 27 s
Channel: Sanchit Gogia
Description: Find the partial fraction decomposition. (2 x^3+2 x^2+4 x-3)/(x^4+x^2)

Find the partial fraction decomposition. (2 x^3+2 x^2+4...

Key Concepts

Factorization of Polynomials

Factorization involves breaking down a polynomial into a product of simpler polynomials, often linear or quadratic factors. This step is crucial in partial fractions because it identifies the distinct components of the denominator that will be used to set up the individual fractional terms.

Handling Repeated and Irreducible Factors

When the denominator of a rational function contains factors that repeat or are irreducible quadratics, each repeated factor must be accounted for with separate terms (with ascending powers) in the decomposition. Similarly, irreducible quadratic factors will necessitate numerators of one degree less than the factor to ensure the decomposition is valid.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational function as a sum of simpler fractions. This technique is particularly useful in integration, solving differential equations, and simplifying algebraic expressions, as it breaks down complicated functions into pieces that are easier to manage.

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. Understanding their structure, including the degrees of the numerator and denominator, as well as any inherent restrictions, is essential for techniques like partial fraction decomposition.

Transcript

00:01 Hello, so here we have to find out the partial fraction decomposition of 2x cube plus 2x square plus 4x minus 3 divided by x square plus x -rays to part 4.

00:22 What you can do is you can rewrite this as 2x cube plus 2x square plus 4x minus 3 divide by x square x square plus one and we can write this in partial fraction function form like this a by x plus b by x square plus cx plus d by x squared plus one now what you need to do is if on the partial fraction decomposition you just move the denominator to the right side and you cancel denominators out and you get something like a x x square plus 1 plus b x square plus 1 plus c x cube plus d x square now to find the values of a b and c okay first all we got to equate them to this to find the values of a b c and d what i can do is we can put the values of x's roots so this there only is one real root in these equations that is zero because x -go plus one cannot be zero with real numbers so put x to the zero and you get the value of b which is going to be this will become zero this will come zero this will come zero so then you get b which is minus 3 now you have the value of b now what you need to find the value of b now what you need to find the value of a, c, and d is you can just combine the powers or group the powers together.

02:38 So let's just say you're going to find the value of a, right? no, wait.

02:45 Let's say you're going to find the value of d.

02:47 So d x squared plus b, okay, since there's only 2x, 2 is the coefficient of x squared here, that means that and bx squared here, that means that bx square plus d x square is equal to 2x square, right? you cancel the x -quered route, that means b plus d is 2.

03:04 That means b is equal to, i mean, that means d is equal to 5.

03:09 You do the same thing for c.

03:13 So there's xq here...

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