5–16. Set up the appropriate form of the partial fraction deco...

Question

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

6. (4x + 1)/(4x² - 1)

Accepted Answer

Although, in this video, we are going to be finding the partial fraction decomposition for the given rational expression, but we are not going to be solving for the unknown constants. So, the rational expression given to us is the quantity 7 Z + 3 divided by Z2 minus 9. Now, in order to approach this kind of problem, the first thing we want to do is we want to make sure and check to see if the denominator is fully factored. Now, in the denominator, we have Z2 minus 9, and we can factor this by using the difference of square terms. By using the difference of square terms, this is going to factor to be Z minus 3, multiplied by Z + 3. And so if we apply this factorization to the denominator, we can rewrite the expression to be 7 Z + 3 divided by the quantities Z minus 3, multiplied by Z + 3. Now, since we have two different factors in the denominator, what this means is that we are going to have 2 partial decompositions or 2 partial fractions. Now, in order to define what kind of partial fractions we have, we have to take a look at both of the quantities in the denominator. If we look at Z minus 3, this is a linear function, which means we will have a linear partial fraction. Furthermore, this quantity is raised to the power of 1, which means it does not repeat, so it is non-repeating. And so what this means is that we can define the first partial fraction to be A divided by Z minus 3. And the same characteristics can be said for the second factor. Z + 3 is a linear non-repeating factor in the denominator, so its partial fraction can be defined as B divided by the quantity C + 3, and this is going to be the partial fraction decomposition to the given rational expression. And with that being said, the overall solution to this problem is going to be a. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
6. (4x + 1)/(4x² - 1)

Key Concepts

Partial Fraction Decomposition

Factoring Quadratic Expressions

Form of Partial Fractions for Distinct Linear Factors

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