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

Question

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

20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says to express the quantity of X2 + 4 X + 5 in quantity, divided by the quantity of the quantity of X + 2 in quantity multiplied by the quantity of X minus 1 in quantity multiplied by the quantity of X minus 4 in quantity as a sum of partial fractions. So, we want to take our fraction here and actually break it up into multiple fractions using partial fractions. So, the quantity of X2 plus or X. Plus 5 in quantity, divided by the quantity of the quantity of X plus 2 in quantity multiplied by the quantity of X minus 1 in quantity, multiplied by the quantity of X minus 4 in quantity is going to be equal to. And so what we're going to do is have a coefficient divided by each one of the factors in the denominator. This is going to be equal to a divided by the quantity of X + 2 in quantity, plus B divided by the quantity of X minus 1 in quantity, plus C divided by the quantity of X minus 4 in quantity. And so we need to solve for the three unknowns A, B, and C. So the first step is to multiply both sides of this equation by the denominator on the left hand side. In doing so, we will have that X2 + 4 X. Plus 5 is equal to a multiplied by the quantity of X minus 1 in quantity multiplied by the quantity of X minus 4 in quantity plus B multiplied by the quantity of X plus 2 in quantity multiplied by the quantity of X minus 4 in quantity. Plus C multiplied by the quantity of X plus 2 in quantity, multiplied by the quantity of X minus 1 in quantity. Now we just need to solve for the coefficients A, B, and C. So to solve for the coefficient of A, we're gonna set X equal to minus 2. And what that's going to do is cause the factor of X + 2 to be equal to 0, which will eliminate our coefficients B and C out of our equation. So setting X equal to -2, we will get that we have the quantity of -2 and quantity squared. Plus or multiplied by minus 2. Plus 5. is equal to a multiplied by the quantity of -2, -1. In quantity multiplied by the quantity of -2 minus 4 in quantity, plus B multiplied by the quantity of minus 2 + 2 in quantity multiplied by the quantity of -2 minus 4 in quantity, plus C multiplied by the quantity of minus 2 + 2 in quantity multiplied by the quantity of -2 minus 1 in quantity. So simplify this expression. We see that we're gonna have. 1 is equal to 18A. And so now we just need to divide both sides of the of this equation by 18 to solve for A. So that means that A is going to be equal to 1 divided by 18. Now, to solve for the coefficient of B, We're going to set X equal to 1. So, substituting N X equal to 1 into our expression, we will have 1 squared. Plus 4 multiplied by 1 + 5 is equal to a multiplied by the quantity of 1 minus 1 and quantity multiplied by the quantity of 1 minus 4 in 20 plus b multiplied by the quantity of 1. Plus 2 in quantity multiplied by the quantity of 1 minus 4 in quantity, plus C multiplied by the quantity of 1 + 2 in quantity multiplied by the quantity of 1 minus 1 in quantity. And notice that the coefficients A and C are multiplied by the quantity of 1 minus 1, which is going to be 0. So, simplifying this expression, we see that we have 10 is equal to minus 9 B. So, define both sides of the equation by minus 9, we see that B is going to be equal to minus 10 divided by 9. And finally, to solve for our coefficient. C, we're gonna set X. Equal to 4. So the quantity of X minus 4 is going to be equal to 0, and that's going to eliminate our coefficients A and B. So, setting X equal to 4, we will have 4 squared Plus 4 multiplied by 4. Plus 5 is equal to a multiplied by the quantity of 4 minus 1 in quantity, multiplied by the quantity of 4 minus 4 in quantity. Plus B multiplied by the quantity of 4 + 2 in quantity multiplied by the quantity of 4 minus 4 in quantity plus c multiplied by the quantity of 4 + 2 in quantity multiplied by the quantity of 4 minus 1 in quantity. And so, simplifying this expression, we see that we will have 37 is equal to. 18, multiplied by C, and solve for C, we'll just divide both sides of the equation by 18. In doing so, we see that C is going to be equal to. 37 divided by 18. That means that the quantity. Of X squared Plus 4 X. Plus 5 in quantity divided by the quantity of the quantity. Of X Last 2. In quantity multiplied by the quantity of X minus 1 in quantity multiplied by the quantity of X minus 4 in quantity is going to be equal to 1 divided by the quantity of 18, multiplied by the quantity of X plus 2 in quantity. Minus 10 divided by the quantity of 9, multiplied by the quantity of X minus 1 in quantity. Plus 37 divided by the quantity of 18, multiplied by the quantity of X minus 4 in quantity. And this here is the answer that we're looking for for this problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

17-22. Give the partial fraction decomposition for the following expressions.20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))

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17-22. Give the partial fraction decomposition for the following expressions.
20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))