In Exercises 9–30, use the Binomial Theorem to expand each binomi...

Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x²+2y)^4

Accepted Answer

Hey everyone here we are asked to perform binomial expansion and simplify the resulting expression for the given binomial where we have the quantity of X squared plus six Y to the power of four. So before we start expanding, let's recall the formula for binomial expansion where we have the quantity of A plus B raised to the nth power is equal to. And now expanding, we have n zero times A. To the power of N plus n. One times A. To the power of n minus one times B plus n. Two times A. To the power of n minus two times B squared plus. And then I'm going to continue below here. We have plus N three times A. To the power of n minus three times B cubed. And then we continue this expansion here until we're left with N over N and times B to the power of N. And then this whole expansion here is equivalent 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 R. And so now with this information we can use this formula for our given binomial. Here again we have the quantity of X squared plus six Y raised to the fourth power. So because our binomial is raised to the fourth power, our N is going to be equal to four. So now keeping the variables as they are in the formula we have the quantity of A plus B to the fourth power is equal to. And now we begin expanding. We have instead of N. Again we replace it with four. So we have +40 times A to the power of four plus and one or sorry not N. We replace N with four. We have +41 times a cubed times B. Then we have plus four, two times A. To the power of two times B to the power of two plus 43 times A times B cubed plus for four. And finally be to the times B to the fourth power. And now we can expand or rewrite this expansion here including the actual coefficient, the binomial coefficients here. So if we recall how to solve for these coefficients, we have the expression here, A to the fourth power plus four times a cubed times B plus six A squared times B squared plus four A. Times B cubed plus four, sorry, plus B. With a coefficient of one to the fourth power. And now with this we just have to substitute in the proper terms for A and B. So looking at our given expression here, our first term, our a term is going to be X squared and then our second term is six Y. So this will be our B. So b is going to be six Y. And now we can go ahead and just substitute in these terms. So again we have the quantity of now instead of A. We have x squared plus B which is six Y again raised to the fourth power is equal to Now. Using our simplified expression from from before we have X squared to the fourth power plus four times a cube. So we're gonna have X square raised to the third power times B which is six Y. Then we have plus six times A. Which is X squared to the second power. So we're going to square the X squared. And then we have times B six Y squared And then we have four times x squared times be cubes. So we have times six Y raised to the third power. And lastly we have plus six Y Raised to the 4th power. And now we can just clean up this expansion just simplifying a bit. The powers here we have the quantity of X squared plus six Y. To the power of four is equal to X to the eighth power plus 24 X to the sixth power times Y plus 206. 16 X to the fourth power times Y squared plus 864 X squared times Y cubed plus times Y to the fourth power. And with this our expansion is simplified and we are left with this expression here which gives us answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x²+2y)^4

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplification of Expressions

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