In Exercises 39–48, find the term indicated in each expansion. (2...

Question

In Exercises 39–48, find the term indicated in each expansion. (2x + y)^6; third term

Accepted Answer

Hey everyone here we are asked to find the fifth term of the expression. The quantity of three D plus two E. To the eighth power using binomial expansion. So if we recall the formula for binomial expansion is the quantity of A plus B to the n power is equal to the sum from R is equal to zero to n. Of n R times A to the power of n minus R, times B to the power of are now looking at our formula, we can see that our first term in our expansion begins when R is equal to zero. So for our fifth term we know that our must be equal to four. And now we want to work to find the rest of the variables from our formula in relation to our given expression. So we can begin with our A term here and looking at our given expression, we can see that A is going to be three D. And that means B is going to be this to E. And lastly our binomial is raised to the eighth power. So we can see that end has the value of eight. And now with this we can just plug in this information into our formula and find our fifth term. So beginning with the coefficient in this set of parentheses, the top value is end. So we have eight and the lower value is our so we have four. Next we have times A. Which is three D. To the power of N -R which is really 8 -4 times B. Which again is to E. To the power of our which is for now we just need to start simplifying and we can begin by calculating the binomial coefficient here. But we need to recall that this form and R. Is equal to N. Factorial divided by the quantity of N minus r. Factorial times R factorial. And now with this we've already identified our N. And our values so we can just go ahead and plug them into this formula here. So our coefficient becomes eight factorial divided by the quantity of eight minus four factorial times four factorial. And then our other terms we can just write them down here. So we have times three D. To the power of eight minus four times to E. To the power of four. And now we can just begin simplifying. So we see beginning with our coefficient here we have eight factorial divided by eight minus four gives us four factorial times four factorial. And then we have the quantity of three D. To the power of eight minus four which just gives us four times the quantity of two E. Two. The power of four. And now we can continue to simplify and find that the fifth term of this expansion here is 90,720 times D. To the power of four times E to the power of four. And with this we are left with answer choice B. Thanks for watching. I hope you found this video helpful. And I'll see you again next time.

In Exercises 39–48, find the term indicated in each expansion. (2x + y)^6; third term

Key Concepts

Binomial Theorem

Binomial Coefficient

Term in a Polynomial Expansion

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