Find the constant term in the expansion of (x + {2}/{x^2})^{15}.

Question:

Find the constant term in the expansion of {eq}\displaystyle \left ( x + \frac{2}{x^2} \right )^{15} {/eq}.

Binomial Expansion:

If an expression is given in the form {eq}(a + b)^n {/eq}, then the given expression can be expand by using the binomial expansion. The following formula used in the binomial expansion:

{eq}(a + b)^n = \,^nC_0 a^nb^0 + \, ^nC_1 a^{n-1}b^1 + \, ^nC_2 a^{n-2}b^2 + ... + \, ^nC_r a^{n-r}b^r + ... + \, ^nC_n a^{n-n}b^n \\ (a + b)^n = a^n + \, ^nC_1 a^{n-1}b + \, ^nC_2 a^{n-2}b^2 + ... + \, ^nC_r a^{n-r}b^r + ... + \, b^n {/eq}

Where, the {eq}r^{th} {/eq} term of the binomial expansion is, {eq}^nC_r a^{n-r}b^r {/eq}.

Answer and Explanation:

The given expression is:

{eq}\left ( x + \dfrac{2}{x^2} \right )^{15} {/eq}

We have,

{eq}a = x, \quad b = \dfrac{2}{x^2}, \quad n = 15 {/eq}

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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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