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. (5x – 1)^3

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 eight y minus one raised to the third power. So before we start expanding here, let's recall the formula for binomial expansion and we have the quantity of A plus B. To the n. Power is equal to 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 A. Sorry, plus n three times A. To the power of n minus three times B cubed. And now continuing here below we have plus and we continue expanding until we reach N. N times B to the power of N. And now this expansion here is equal to the sum of 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. And now with this formula here we can go ahead and use it for our expansion of the given expression which again is the quantity of eight Y minus one to the power of three. So in this case our binomial is raised to the third power so our N is going to be three. And so now we can write out our expansion keeping the A. And B variables for now but replacing N with three. So we have the quantity of A plus B to the third power is equal to again replacing N with three. We have +30 times a. Cute plus three. One times A. To the power of two times B. Then plus three two times a times B squared plus three three times B cubed. And now with this we just want to simplify this expansion, writing out the actual coefficients for the binomial is here. So if we recall solving for binomial co coefficients, we are left with the expansion of a cubed plus three A squared times B plus three A times B squared plus B cubed. And now with this we just need to substitute in the appropriate terms for A and B in relation to our given expression. So according to our formula A is the left hand side term. So A. Is A Y. And then for B we have this in the formula formula we have plus B. But in our expression we have minus one. So our B is just going to be negative one. And so now with this we just need to substitute in these terms into our current expansion. So again we have the quantity of a plus B cubed but instead of a plus B we have eight y Plus -1. So we just have eight Y -1. And now we can start expanding. So again we have a cube. So we have eight y raised to the third power plus three times eight Y squared times B which again is negative one plus three times 8 y times B squared. So we have negative one squared Plus our last term here. We will have negative one. Cute. And now we just have to simplify this expansion here. So we have the quantity of eight Y minus one to the third power is equal to 512 Y. Cute minus 192 Y squared plus 24 y minus one. And with this we are done simplifying our binomial expansion. And so 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.

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In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (5x – 1)^3