Partial Fraction Decomposition Find the partial fraction decomposition of the rational function.

Partial Fraction Decomposition Find the partial fraction...

Key Concepts

Rational Functions

A rational function is any function that can be written as the quotient of two polynomials. Understanding the behavior and structure of rational functions is essential for many areas of mathematics, including integration and the analysis of asymptotic behavior.

Polynomial Factorization

Factoring polynomials involves expressing a polynomial as a product of simpler polynomials. This is particularly important in partial fraction decomposition because it allows one to break down the denominator into factors, which then guides the form of the decomposition.

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 especially useful in calculus for integrating rational functions and in other areas where simpler terms are easier to handle.

Decomposition with Repeated Factors

When the denominator of a rational function contains repeated factors, the structure of the partial fractions must be adjusted accordingly to account for the multiplicity. Each repeated factor will contribute multiple terms in the partial fraction expansion, each with increasing powers in the denominator.

Transcript

00:01 So we're being asked to do the partial fraction decomposition for our given fraction.

00:05 Well, the first thing we must do is factored the denominator.

00:09 Well, when we go to a factor 4x squared plus 12x plus 9, you'll find that that is a perfect trinomial.

00:15 And when we factor it, what we're going to get is the quantity of 2x plus 3 squared.

00:23 And now that our fraction or our denominator is factored, we can now set up our partial fraction decomposition.

00:29 Well, if you notice that this factor, 2x plus 3, is repeated twice.

00:35 So to do our partial fraction decomposition, we're going to have a over 2x plus 3 plus b over the quantity of 2x plus 3 squared.

00:48 But we know that this is going to equal to 2x over the quantity of 2x plus 3 squared.

00:55 So now we've written our partial fraction decomposition.

00:58 So now what i'm going to do is i'm going to multiply both sides of our equation by our common denominator, which is the quantity of 2x plus 3 squared.

01:08 And the reason why we do this is so that way on the right hand side, it will cancel out our denominator, and we're just left with 2x.

01:18 So now, on the left hand side, our first fraction was a divided by 2x plus 3.

01:23 So it has that factor 2x plus 3 once, but it needs it again.

01:27 So i need to multiply a by 2x plus 3.

01:32 Now for my second fraction, which is b over 2x plus 3, what i find is that i already have that 2x plus 3 squared in the denominator, so i just have my b term.

01:44 So now what i'm going to do is i'm going to distribute that a term...

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