17-22. Give the partial fraction decomposition for the followi...

Question

17-22. Give the partial fraction decomposition for the following expressions.

17. (5x - 7) / (x² - 3x + 2)

Accepted Answer

Hello, in this video, we want to determine the partial fraction decomposition for the given rational expression. The expression given to us is 3 X + 5 divided by the quantity X2 minus 4 X + 3. Now, in order to find the partial fraction decomposition for this, the first thing we want to do is you want to fully simplify our denominator. Now, our denominator is in the form of a factorable trinomial. X2 minus 4 X + 3 will factor to be X minus 3 multiplied by X minus 1. So we can rewrite the expression to be 3 X + 5 divided by the quantity. X minus 3 Divide multiplied by x minus 1. Now, in this case here, if we take a look at the two quantities, the two factors in the denominator, we have non-repeating linear factors. Since we have non-repeating linear factors, we will go ahead and create a partial fraction for each of those linear factors. So, for the first linear factor X minus 3, the first partial fraction is going to be defined as a divided by X minus 3. Then we will add the second partial fraction for the non-repeating linear factor of X minus 1, and that is going to be defined as B divided by X minus 1. Now, what we need to do in this case is we need to go ahead and solve for the coefficients A and B. Now, in order to do this, we will first need to multiply the entire equation by the lowest common denominator, and the lowest common denominator in this case is going to be the multiples of the different factors in the denominators. That is going to be X minus 3 multiplied by X minus 1. And so if we multiply the entire equation by this quantity, we will be left with 3 X + 5. equal to A multiplied by X minus 1, plus B multiplied by X minus 3. Now, what we are going to do is we are going to expand the right hand side of this equation. We will factor A and B into the respective factors, and that is going to leave us with AX minus A, plus B X minus 3 B. Now what we are going to do is we are going to group up the X terms and the constant terms together, and that is going to leave us with A plus B multiplied by X plus the constant group negative A minus 3 B. And keep in mind this is still equal to 3 X plus 5. Now, what we are going to do is we are going to create a system of linear equations in order to solve for both A and B. In order to do this, we are going to set the coefficients of the X terms on both the left and right hand side equal to each other. So, that is going to give us the first linear equation, and that is going to give us A plus B is equal to 3. Then if we were to set the constant terms equal to each other, we get a second linear equation, and that is going to be negative A. -3B is equal to 5. So this is going to be a system of linear equations that will allow us to solve for the coefficients of A and B. Now, what we could go ahead and do in this case is we can add these linear equations together in order to solve for the B term. If we add these equations together, we can cancel out the A terms, and B minus 3B is going to leave us with -2B. And 3 + 5 is going to give us the value of 8. And by dividing by -2, B is equal to -4, which is going to be the first coefficient that we are looking for. Now, if we were to take this B value and plug it back into the first linear equation, we will be left with a minus 4 is equal to 3. And by adding 4 to the right hand side of this equation, we will be left with A is equal to 7, which is going to be the second coefficient that we are looking for. Now that we've solved for the A and B coefficient for the original partial fraction decomposition, by plugging in these coefficients, the partial fraction decomposition of the original rational expression is going to be 7 divided by X minus 3, minus 4 divided by X minus 1, and this is going to be the solution to the given problem. 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 find the partial fraction decomposition, and we will go ahead and see you all in the next video.

17-22. Give the partial fraction decomposition for the following expressions.
17. (5x - 7) / (x² - 3x + 2)

Key Concepts

Partial Fraction Decomposition

Factoring Quadratic Expressions

Setting Up and Solving Systems of Equations

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