Add or subtract, as indicated. 4/x + 1 + 1/x^2 - x + 1 - 12/x^3 +...

Question

Add or subtract, as indicated. 4/x + 1 + 1/x^2 - x + 1 - 12/x^3 + 1

Accepted Answer

Hey everyone here we are asked to perform the following operations and simplify the answer of the given expression of one divided by a plus five plus seven divided by a squared minus five A plus minus seven A plus 35. All divided by a cute plus 25. Now our first step here in being able to perform the necessary addition and subtraction operations is we need to get a common denominator between each of our fractions here in the given expression so we can take a look at our denominators that we have here. And we see from our first fraction we have a plus five from our second fraction we have a squared minus five A plus 25. And from our third fraction we have a cubed plus 125. So now taking a look at these denominators, we can actually identify a pattern between these denominators and that is the sum of two cubes. These denominators follow the pattern of the sum of two cubes. So we need to recall, call the formula for the sum of two cubes which tells us that X cubed plus y cubed is equivalent to the quantity of X plus y times the quantity of x squared minus x, Y plus y squared. So now we need to move on to finding the least common denominator between these denominators that we have and we see that it will actually be our last denominator here which is a cube plus 125. And we can see that this expression matches the left hand side of the sum of two cubes because it is the sum of two cubes. So now we just need to take our least common multiple and divide each denominator by or divide this least common multiple by each denominator so that we know what to multiply each fraction by to get a common denominator. So we take our least common multiple which is a cute plus 125. And we can begin by dividing our first fraction. So we have our least common our first denominator. So we have our at least common multiple divided by eight plus five. And now simplifying, we see that we will have to multiply the numerator and the denominator of our first fraction by a squared minus five A plus 25. And now moving on to our second denominator, we again take the least common multiple of a cubed plus 125 and we divided by a squared minus five A plus 25. And simplifying. We see that we will have to multiply the numerator and the denominator by a plus five. And so now we can go ahead and look back at our original expression and multiply and start finding our common denominators. So beginning with our first fraction, we had one divided by a plus five. Well we know from earlier that we can multiply the numerator and the denominator by a squared minus five plus 25. And again this is for both the numerator and the denominator and now we can move on to our second fraction which is we have plus seven divided by a squared minus five A plus 25. And again we multiply the numerator and the denominator by a plus five. And now for our last fraction we see that we already have the common denominator so we do not have to make any changes. So we just have minus seven A plus 35 all divided by a cubed plus 125. So now we can start simplifying our products here by utilizing our sum of two cubes formula from earlier. So we see that our numerator from our first fraction becomes a squared minus five A plus 25. All divided by a cubed plus 125. And then for our second fraction we get seven times the quantity of a plus five, all divided by a cubed plus 125. And again our last fraction, it's the same. So we have minus seven A plus 35 all divided by a cubed plus 125. So now from here we just have to continue to simplify our expression and now we see that we have common denominators between each of our fractions so we can go ahead and perform the operations as indicated. So again we need to combine our new readers. So doing this, we have a squared minus five A plus 25 from our first numerator. And now factoring in the second from the seven from our second numerator we have plus seven A plus 35 then we have minus the quantity of seven A plus 35. And this is all divided by the common denominator of eight, cubed plus 125. And so now we just need to continue to simplify by now combining our like terms. So again in our numerator combining our like terms, we see that we have seven A plus 35 that cancels out with this negative quantity of seven A plus 35. And so in the numerator we are left with a squared mine five a plus 25. All divided by a cubed plus 125. And now recalling our sum of two cubes formula from earlier, we see that we can expand our denominator. So rewriting our expression with an expanded denominator, we have a squared minus five A plus 25. All divided by the quantity of a plus five times the quantity of a squared minus five A plus 25. And now we see that the entire expression in our numerator cancels with the second set of parentheses. In our denominator. This a squared minus five A plus 25. These entire cancel. So now we can rewrite our expression with what we have left and we see that all we have left is just one divided by the quantity of a plus five. And so this is our final simplified answer one divided by a plus five, which leaves us with answer choice B. Thanks for watching. I hope you found this video helpful, and I'll see you again next time.

Add or subtract, as indicated. 4/x + 1 + 1/x^2 - x + 1 - 12/x^3 + 1

Key Concepts

Rational Expressions

Common Denominator

Polynomial Simplification

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