Chapter 12.5, Problem 4AYU

The ______ ________ can be used to expand expressions like ${\left(2x+3\right)}^6$.

To determine

To fill: The term that is used to refer a expressions like ${\left(2x + 3\right)}^6$.

Answer to Problem 4AYU

Solution:

The Binomial theorem can be used to expand expressions like ${\left(2x + 3\right)}^6$.

Explanation of Solution

Given:

It is asked to fill in the term of a expressions like ${\left(2x + 3\right)}^6$.

Calculation:

The Binomial Theorem:

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

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

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

As the given expression is of the form ${\left(x + a\right)}^n$ with $x = 2x$, $a = 3$ and $n = 6$, it can expand using Binomial theorem.