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.

12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

Accepted Answer

Hello. In this video, we are going to be finding the partial fraction decomposition for the given rational expression, but we are not going to solve for the unknown constants. So, the rational expression given to us is 5 X2 + 7 divided by the quantity X2 minus 6 X + 9 multiplied by the quantity X2 + 2 X + 5. Now, in order to find the partial fraction decomposition of this, the first thing we want to do is we want to go ahead and make sure that our denominator is fully factored. Now, if we take a look at the first quantity, we can factor X2 minus 6 X + 9. This quantity is going to factor to be X minus 3 quantity squared. Now, X2 + 2 X + 5 is not in a factorable form, so we're going to leave it as it is. So we can rewrite the rational expression as 5 X2 + 7 divided by the quantity, X minus 3 quantity squared multiplied by X2 + 2 X + 5. Now, since we have a fully factored denominator, we can go ahead and start to set up the partial fraction decomposition. Now, let's go ahead and take a look at the first quantity, X minus 3 quantity squared. Now, if we ignore the exponent for now and just pay attention to X minus 3, the type of function that this is is a linear function. And because we have an exponent of 2, this is telling us that we have 2 instances of X minus 3, and that means that X minus 3 quantity squared is going to identify as a linear repeating function. So what this means is that if we want to create the partial fractions for this quantity, we are going to have to create linear partial fractions that lead up to X minus 3 quantity squared. So the first one is going to be defined as a divided by 1 instance of X minus 3. Then we will add one more linear function. Be divided by X minus 3 quantity squared, and this is where we will stop for the linear repeating function. Now let's take a look at the second factor in the denominator, that is X2 + 2 X + 5. Now, this is a non-repeating function, so this does not repeat mean that we will only have one partial fraction for this factor in the denominator. But notice that the leading coefficient is 2. What this means is that this type of function is going to identify as a quadratic function. And so what this means is that we are going to make a quadratic non-repeating partial fraction for this factor in the denominator. So that is going to be defined as CX plus D divided by X2 + 2 X + 5. And so with that being said, this is going to be the partial fraction decomposition for the original rational expression. And with that being said, the overall solution to this problem is going to be B. So I hope this video helps you in understanding how to find partial fractions, 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.
12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

Key Concepts

Partial Fraction Decomposition

Factoring Quadratic Expressions

Form of Partial Fractions for Quadratic Denominators

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