In this lesson, our instructor Vincent Selhorst-Jones teaches about Partial Fractions. Youll begin with the prerequisites and uses, and then learn about proper and improper polynomial fractions. Vincent also goes over linear factors, irreducible quadratic factors, mixing factor types, and how to figure out the numerator. The lesson ends with four examples to give you extra practice.
Note:Before starting, it should be noted that this is a rather difficult concept to explain just with writing. It helps a lot to see it in action, so it's strongly recommended that you watch the video if you find any of this confusing.
Long ago, you learned how to combine fractions: you put them over a common denominator, then combine them. But what if we wanted to do the reverse? What if we had a fraction that we wanted to break up into multiple, smaller fractions? We call these smaller
fractions partial fractions and the process partial fraction decomposition.
To understand this lesson, you'll need some familiarity with solving systems of linear equations. Previous experience from past Algebra classes will probably be enough, but if you want a refresher, check out the lesson Systems of Linear Equations.
Knowing how to factor polynomials and understanding them in general will also be necessary, along with the ability to do polynomial division.
If we have a polynomial fraction of the from [N(x)/D(x)], there are two possibilities in regards to the degrees of the top and bottom polynomials:
Proper: degree of N(x) < degree of D(x);
Improper: degree of N(x) ≥ degree of D(x).
To decompose a polynomial fraction, it must first be proper. If the fraction is improper and we want to decompose it, it must first be made proper through polynomial division. [Remember that the remainder from the division goes back on the
original denominator, which we will then decompose.]
Once the polynomial fraction is proper, the next step is to factor the denominator. After it's broken down into its smallest factors, we're ready to decompose. Notice that there are two types that these smallest possible factors can come in:
Linear factors raised to a power: (ax+b)m;
Irreducible quadratic factors raised to a power: (ax2+bx+c)m. [Remember, `irreducible' means it can't be broken up further. That means quadratics like x2+1 or 5x2−3x+20.]
The partial fraction decomposition must include the following for each linear factor (ax+b)m:
where A1, …, Am are all constant real numbers.
The partial fraction decomposition must include the following for each irreducible quadratic (ax2+bx+c)m:
where A1, …, Am and B1, …, Bm are all constants.
If we have multiple of a given type or mixed types, we just decompose each of them based on their rules and put them all together.
Finally, we need to figure out what goes on the numerators. To do this, have the original fraction on one side of an equation and the partial fraction decomposition on the other. Then multiply both sides of the equation by the denominator of the original
fraction. This gives an equation that we can transform into a system of linear equations and solve to find the values for each of the constants.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.