Use the Binomial Theorem to expand each binomial and express t...

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x+2)³

Accepted Answer

Hey everyone here we are asked to perform binomial expansion and simplify the resulting expression for the following binomial where we have the quantity of X plus seven cubed. So here before we begin expanding we need to recall the formula for binomial expansion where we have the quantity of A plus B to the end. Power is equal to now the expansion and zero times A. Two. 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 n three times A. To the power of n minus three times B cubed. And then we continue expanding here all the way until we are left with an and times B to the power of N. And now this entire expansion is the same as taking 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 our. So with this formula in mind we can use it to stay are expanding and were given binomial expression which is the quantity of X plus seven cubed. And now since our binomial is cubed we know that N is going to be replaced by three in our formula. So doing this we can start writing our expansion out. But keeping the A. And B variables from the formula we have the quantity of A plus B cubed is equal to +30 times A. To the power of three plus 31 times A. To the power of two times B plus three, two times a times B squared plus our last term. +33 times be cute. And now with this we just need to recall on how to calculate the coefficients for a binomial. And with this we can write our simplified expansion as a cute plus three times a squared times B plus three times a times B squared plus B. Cute. And now with this all that's left to do is substitute in the appropriate terms for A. And B. So looking at our given binomial we see that the left hand side term is the A value. So A. Is just going to be X. And then our other term is positive seven. So B is just going to be seven. And now with this all we have to do is plug in these values for A. And B. So we see that the quantity of A which is X plus B which is seven cubed is equal to the expansion here. Where we have X cubed Plus three times x squared times seven plus three times X plus or sorry not plus times we have three X times seven squared plus seven cubed. And now all that is left to do is simplify our expression here. So we see that the expansion of the binomial, the quantity of X plus seven cubed. The expansion is equal to X cubed plus 21 X squared plus 147 X plus 343. And with this we are done finding our expansion and simplifying and we are left with answer choice D. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x+2)³

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplifying Polynomial Expressions

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