Chapter 12.5, Problem 39AYU

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.

The coefficient of $x^0$ in the expansion of ${\left(x^2+\frac{1}{x}\right)}^{12}$

To determine

To find: The coefficient of $x^0$ in the expansion of ${\left(x^2 + \frac{1}{x}\right)}^{12}$ using Binomial Theorem.

Answer to Problem 39AYU

Solution:

The coefficient of $x^0$ in the expansion is 495.

Explanation of Solution

Given:

Expression is given as ${\left(x^2 + \frac{1}{x}\right)}^{12}$

Formula used:

The Binomial Theorem:

Let $x$ and $a$ be real numbers. For any positive integer $n$, we have

$\begin{array}{l}{\left(x + a\right)}^n = \left(\begin{array}{c}n\\ 0\end{array}\right)x^n + \left(\begin{array}{c}n\\ 1\end{array}\right)a{x}^{n - 1} + \cdots + \left(\begin{array}{c}n\\ j\end{array}\right)a^j{x}^{n - j} + \cdots + \left(\begin{array}{c}n\\ n\end{array}\right)a^n\\ {\left(x + a\right)}^n = {\displaystyle \sum _{k = 0}^n\left(\begin{array}{c}n\\ k\end{array}\right)x^{n - j}{a}^j}\end{array}$

The binomial theorem can be used to find a particular term in an expansion without writing the entire expansion.

Based on the expansion of ${\left(x + a\right)}^n$, the term containing $x^j$ is $\left(\begin{array}{c}n\\ n - j\end{array}\right)a^{n - j}{x}^j$.

Here $x = x^2,a = \frac{1}{x}$ and $n = 12$. The only time we will be left with an $x^0$ term is when $j = 4$,

$\left(\begin{array}{c}12\\ 12 - 4\end{array}\right){\left(\frac{1}{x}\right)}^{12 - 4}{\left(x^2\right)}^4 = \left(\begin{array}{c}12\\ 8\end{array}\right){\left(\frac{1}{x}\right)}^8\cdot \left(x^8\right) = 495$

The coefficient of $x^0$ in the expansion is 495.