Find the partial fraction decomposition. (x^3)/(x^3-3 x^2+9...

Key Concepts

Partial Fraction Decomposition

This is a technique used to express a rational function as a sum of simpler fractions, which is particularly useful in algebra and calculus for integration and inverse Laplace transforms. It involves breaking down a complex fraction into a series of terms with simpler denominators that are usually linear or quadratic factors.

Polynomial Division

When the degree of the numerator is greater than or equal to the degree of the denominator, polynomial division is used first to rewrite the rational function as the sum of a polynomial and a proper rational function, where the degree of the numerator is less than the denominator. This step simplifies the process of applying partial fraction decomposition.

Factorization of Polynomials

Factorization is the process of writing a polynomial as a product of its factors. In the context of partial fractions, factorizing the denominator into irreducible factors (linear or quadratic) is crucial because the form of the partial fractions depends on these factors, and it determines the coefficients to be determined in the decomposition process.

Rational Functions

A rational function is a ratio of two polynomials. Understanding their structure, including degrees of the numerator and denominator and the factorization of the denominator, is fundamental to applying techniques such as polynomial division and partial fraction decomposition effectively.

Transcript

00:01 We have to find out the partial fraction decomposition of x -cube divided by x -cube minus 3x -square plus 9x minus 27.

00:26 So we see that the degrees are, well, yeah, they're equal, right? we need it to be smaller.

00:33 So what we do is we do long division of x -cube by this.

00:37 And when we do that, we get the answer as 1 plus.

00:44 3x square minus 9x plus 27 by x cube minus 3x square plus 9x minus 27 so now we can rewrite this whole thing as x square plus 9 to add by x minus 2 now to do that now to find the partial fraction what we'll do is you'll just take this fraction and rewrite this as 3x square minus 9 x plus 27 by x squared plus 9 x minus 3 is equal to a by a x squared plus 9 plus c by x minus 3.

02:00 Now we have to find the values of a b and c so what we'll do is we'll take the denominator to the right side to cancel the nominaries out since the partial fraction we have to find out you get something like 3x square minus 9x plus 27 is equal to ax x minus 3 plus b x minus 3 plus c x squared plus 9.

02:34 To find the values of a, b, and c, what it will do is we will plug in the values of x at its roots.

02:40 So there's one root here that's in that's in that denominator.

02:44 That would be x is equal to three right once you would x equal to three we get is a is going to be zero b is going to be zero so you get c so that means nine plus nine which is uh 18 so that means 18 c is equal to 27 minus 27 plus 27 that means c is equal to three by two now we have to find the values of a and b, right? so to do that, what we'll do is we'll take the, to do this, all we'll do is we'll group of the powers of x together...

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