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Which term of the expansion of (x^{4} + 5)^{10} contains x^{16}?

Question:

Which term of the expansion of {eq}\left(x^{4} + 5\right)^{10} {/eq} contains {eq}x^{16} {/eq}?

Binomial Expansion:

The binomial expansion of {eq}(x+y)^n = nC0 x^n y^0 +nC1 x^{n-1} y^1 +nC2 x^{n-2} y^2 +nC3 x^{n-3} y^3 + \cdots +(-1)^r nCr x^{n-r} y^r+\cdots {/eq}. The binomial theorem is used in many application in mathematics.

The general term is of the from {eq}T_{r+1}=(-1)^r nCr x^{n-r} y^r {/eq}.

Answer and Explanation:

We are given {eq}(x^{4} + 5)^{10} {/eq}


The general term of {eq}(x+y)^n {/eq} is {eq}T_{r+1}=nCr x^{n-r} y^r. {/eq}


In this problem, {eq}x=x^4 . y = 5 , n=10 , r = 6 {/eq}

{eq}T_{4+1}= 10C6 (x^4)^{10-6} (5)^6 {/eq}

{eq}\Rightarrow T_{5}= 210 (x^4)^{4} \cdot 15625 {/eq}

{eq}\Rightarrow T_{5}=3281250 x^{16} {/eq}


Therefore, the numerical coefficient of the {eq}x^{16} {/eq} term in the expansion is {eq}=3281250 {/eq}.


Learn more about this topic:

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Binomial: Definition & Examples

from High School Algebra I: Help and Review

Chapter 24 / Lesson 5
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