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Chapter 14: Problem 19

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(y-3)^{4}$$

Short Answer

Expert verified
(y-3)^4 simplifies to \(y^4 - 12y^3 + 54y^2 - 108y + 81\).

Step by step solution

01

Identify the Terms

Since we are given to expand \( (y-3)^{4}\) using Binomial Theorem, here y is 'a', -3 is 'b', and 4 is 'n'.
02

Substitute the Values into the Formula

Applying the Binomial Theorem, we obtain \[ (y-3)^4 = \sum_{k=0}^{4} {4 \choose k} y^{4-k} (-3)^{k}\]
03

Apply binomial coefficients and calculate each term

Let's calculate it term by term: \[ - {4 \choose 0} y^{4-0} (-3)^{0} = 1*y^4*1 = y^4, - {4 \choose 1} y^{4-1} (-3)^{1} = 4*y^3*(-3) = -12y^3, - {4 \choose 2} y^{4-2} (-3)^{2} = 6*y^2*9 = 54y^2, - {4 \choose 3} y^{4-3} (-3)^{3} = 4*y*(-27) = -108y, - {4 \choose 4} y^{4-4} (-3)^{4} = 1*1*81 = 81\]
04

Final Step: Sum Up the Terms

Add all terms together we get: \[ (y-3)^4 = y^4 - 12y^3 + 54y^2 - 108y + 81 \]

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