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How to Perform Partial Fraction Decomposition or Expansion. This method is used to decompose a given rational expression into simpler fractions. In other words, if I am given a single complicated fraction, my goal is to break it down into a series of “smaller” components or parts.
Dec 21, 2020 · Example \(\PageIndex{2}\): Decomposing into partial fractions. Perform the partial fraction decomposition of \(\frac{1}{x^2-1}\). Solution. The denominator factors into two linear terms: \(x^2-1 = (x-1)(x+1)\). Thus $$\frac{1}{x^2-1} = \frac{A}{x-1} + \frac{B}{x+1}.\] To solve for \(A\) and \(B\), first multiply through by \(x^2-1 = (x-1)(x+1)\):
Nov 16, 2022 · The first step is to factor the denominator as much as possible and get the form of the partial fraction decomposition. Doing this gives, \[\frac{{3x + 11}}{{\left( {x - 3} \right)\left( {x + 2} \right)}}\, = \frac{A}{{x - 3}} + \frac{B}{{x + 2}}\] The next step is to actually add the right side back up.
Step 1: Factor the bottom: 5x−4 x2−x−2 = 5x−4 (x−2) (x+1) Step 2: Write one partial fraction for each of those factors: 5x−4 (x−2) (x+1) = A1 x−2 + A2 x+1. Step 3: Multiply through by the bottom so we no longer have fractions: 5x−4 = A 1 (x+1) + A 2 (x−2) Step 4: Now find the constants A 1 and A 2: Substituting the roots, or ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
Sal explains what partial fraction expansion is by rewriting (x²-2x-37)/(x²-3x-40) as the sum of 1 and two rational expressions with linear denominators. Created by Sal Khan.
Partial fraction decomposition is the process of taking a rational expression (that is, a polynomial fraction) and splitting it up (that is, decomposing it) into simpler fractions (being the partial fractions) that, when added, result in the original expression.
