Write the partial fraction decomposition of each rational expression. (-x^2+3 x-9)/(x^3-3 x^2)

Write the partial fraction decomposition of each rational...

Key Concepts

Equating Coefficients

This is a method used to determine the constants in the decomposed partial fractions. By expanding the assumed partial fraction form and matching the coefficients of corresponding powers of the variable on both sides of the equation, the unknown constants can be systematically solved.

Improper Rational Functions

An improper rational function occurs when the degree of the numerator is equal to or greater than the degree of the denominator. In such cases, polynomial long division is used to rewrite the function as the sum of a polynomial and a proper rational function before proceeding with partial fraction decomposition.

Partial Fraction Decomposition

This is a technique used to express a complex rational expression as a sum of simpler rational expressions. It is particularly useful in integration and solving differential equations, as it breaks down a fraction into a set of more manageable parts based on simpler denominators.

Polynomial Factorization

This process involves rewriting polynomials as a product of their factors. In the context of partial fractions, factoring the denominator into linear or irreducible quadratic factors is essential to determine the correct form for the individual partial fractions.

Transcript

00:03 We want to find the partial fraction decomposition of negative x squared plus 3x minus 9 divided by x cubed minus 3x squared.

00:14 So the first thing we want to do is to factor our denominator, which is x cubed minus 3x squared.

00:22 So this can factor into x squared multiplied by x minus 3.

00:27 Now remember that our factor, which is x squared, can be broken into two parts, x to the first, and x squared.

00:38 So we really have three factors in total.

00:42 So in our second step, we want to use these three factors to write our partial fraction.

00:48 So our partial fraction becomes a divided by the first factor, x to the first, plus b divided by the second factor, x squared, and then plus c divided by the third factor, which is x minus 3.

01:04 So this is our partial fraction.

01:07 In our third step, we want to multiply our partial fraction through by the common denominator, which is x squared multiplied by x minus 3, and we want to set this equal to our starting numerator, negative x squared plus 3x minus 9.

01:27 So when we do this, we get that negative x squared plus 3x minus 9 is equal to a multiplied by x times x minus 3 plus b multiplied by x minus 3 plus c multiplied by x squared.

01:57 Now we can simplify this a bit further, so we get negative x squared plus plus x squared plus c multiplied by x squared.

02:03 3x minus 9 is equal to a times x squared minus 3a times x plus b times x minus 3 times b minus 3 times b plus c times squared now we know that the coefficients well actually after this step we want to group the terms the x terms and the constant terms together on the right hand side of our equation so when we group these terms together, we keep our left -hand side the same, so negative x squared plus 3x minus 9, and now we want to group together the x terms and the constant terms.

02:48 So let's first group together our x squared terms...

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