Expand the quotients in Exercises 1–8 by partial fractions.

Question

(2x + 2) / (x² - 2x + 1)

Accepted Answer

Welcome back everyone. Expand 3 x + 5 divided by x2 + 6 x + 9 by partial fractions. For this problem, let's rewrite the fraction. We have 3 x plus 5 divided by x 2 + 6 x + 9. Notice that in the denominator we have a perfect square, so we can rewrite it as follows 3 x plus 5 divided by x plus 3. Squared In this case, we have a repeated linear factor, so we must create a fraction for each power of that factor starting from one up to the highest power, right? So first of all, we're going to write the coefficient a divided by. X + 3 and we're going to add a coefficient b divided by the square of x + 3. Because that's the highest power, right? X plus 3 2. And now we want to identify A and B. So what we're going to do is identify the common denominator in reverse. We have to multiply. The first fraction by x + 3 and divided by x + 3, which gives us a x + 3. Plus b all divided by the common denominator x + 3 squared and we know that this is equal to 3 x plus 5 divided by X + 32 So now let's apply the distributive property in the numerator. We get a multiplied by x + 3a plus b. And we can equate the numerators because we have the same denominators, so this sum must be equal to 3 x plus 5. Now let's set up a system of equations. Notice that on the left hand side, we only have x adjacent to a, and this term must be equal to 3 x, right? So a. Must be equal to 3. We can get the value of a immediately, and the remaining terms are 3 a plus b. Those are the constants. This sum must be equal to the free coefficient of 5. So if a is 3, we can show that from the second equation b is equal to 5 minus 3a. And we can substitute the value of a. So that's 5 minus 3 multiplied by 3, which is equal to 5 minus 9, and that's -4. So the final answer is going to be a divided by x plus 3, which is 3 divided by x + 3 plus b, which is -4. So we can write -4 divided by x + 32. This is how we get the final answer for this problem. Thank you for watching.

Expand the quotients in Exercises 1–8 by partial fractions.(2x + 2) / (x² - 2x + 1)

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Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)