Find the indicated terms in the expansion of the given binomial. The first four terms in the...

Question:

Find the indicated terms in the expansion of the given binomial.

The first four terms in the expansion of {eq}(x^{1/2} + 1)^{40} {/eq}.

Algebraic Expansion of Higher Power:


The required first non-zero terms of the given expression are obtained by the general formula shown below:

{eq}\displaystyle (1+x)^m=1+mx+\frac{m(m-1)}{2!}x^2+\frac{m(m-1)(m-2)}{3!}x^3+\dots {/eq}, where,

  • m is the positive higher power of the algebraic expression.

Answer and Explanation:


The given algebriac expression is:

{eq}f(x)=(x^{1/2} + 1)^{40}\\ =( 1+x^{1/2})^{40} {/eq}

After the comparison of the above expression with the general algebraic expression {eq}(1+x)^m {/eq}, we have:

{eq}m=40\\ x=x^{1/2} {/eq}


By the formula of the series expansion and the above values, we have:

{eq}\begin{align*} \displaystyle ( 1+x^{1/2})^{40}&=1+40x^{1/2}+\frac{40(40-1)}{2!}(x^{1/2})^2+\frac{40(40-1)(40-2)}{3!}(x^{1/2})^3+\dots\\ &=\displaystyle 1+40x^{1/2}+\frac{40(39)}{2}(x^{2/2})+\frac{40(39)(38)}{3(2)1}(x^{3/2})+\dots\\ &=\displaystyle 1+40x^{1/2}+780x+9880x^{3/2}+\dots\\ \end{align*} {/eq}


Learn more about this topic:

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How to Use the Binomial Theorem to Expand a Binomial

from Algebra II Textbook

Chapter 21 / Lesson 16
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