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Chapter 14: Problem 20

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

Short Answer

Expert verified
The simplified form of the binomial expansion (y-4)^4 is \(y^4 - 16y^3 + 96y^2 - 256y + 256\)

Step by step solution

01

Introduction of the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here \(a = y\), \(b = -4\), and \(n = 4\). Therefore, we apply these values in the Binomial Theorem.
02

Expansion and exponents simplification

We get \((y - 4)^4 = \sum_{k=0}^{4} \binom{4}{k} y^{4-k} (-4)^k\) = \(\binom{4}{0} y^4 (-4)^0 + \binom{4}{1} y^{3} (-4)^1 + \binom{4}{2} y^{2} (-4)^2 + \binom{4}{3} y^1 (-4)^3 + \binom{4}{4} y^{0} (-4)^4\)
03

Use the formula for binomial coefficients and simplify

Using the formula for binomial coefficients, \(\binom{n}{r} = n! / [(n - r)!r!]\), and simplfying each term gives us: \(1*y^4*1 - 4*y^3*4 + 6*y^2*16 - 4*y*64 + 1*256\)
04

Calculate the coefficient for each term

Note that the powers of -4 increase with each term based on the index k, and taking into account the final sign and simplifying we get: \(y^4 - 16y^3 + 96y^2 - 256y + 256\)

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Consider the sequence \(8,3,-2,-7,-12, \ldots .\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Use the formula for the general term (the nth term) of a geometric sequence to solve. Suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the thirtieth day of the month?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=0}^{6}(-1)^{i}(i+1)^{2}=\sum_{i=1}^{7}(-1) j^{2}$$

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 69 and 70 to verify the expansion. $$f_{1}(x)=(x+2)^{6}$$
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