Chapter 4.2, Problem 44E

To determine

To expand: The Binomial ${\left(6m+2\right)}^2$ by using the Pascal Triangle.

Answer to Problem 44E

The expansion of the expression ${\left(6m+2\right)}^2$ is $36m^2+24m+4$ .

Explanation of Solution

Given information: The expression is ${\left(6m+2\right)}^2$ .

Theorem used: The Binomial Theorem: ${\left(x+y\right)}^n=x^n+n{x}^{n-1}y+\cdots {+}_n{C}_r{x}^{n-r}{y}^r+\cdots +nx{y}^{n-1}+y^n$ , the coefficient of $x^{n-r}{y}^r$ is ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}$ .

Calculation:

Here, $n=2$ .

First write out the Pascal’s triangle such that the row that begins with 1, 2.

  $\begin{array}{ccccccccccc}& & & & & 1& & & & & \\ & & & & 1& & 1& & & & \\ & & & 1& & 2& & 1& & & \\ & & 1& & 3& & 3& & 1& & \\ & 1& & 4& & 6& & 4& & 1& \\ & & & & & & & & & & \end{array}$

From the Pascal’s triangle, it is observed that the binomial coefficients of the expression are 1,2, and 1.

Expand the expression ${\left(6m+2\right)}^2$ by using formula in Binomial theorem as follows.

  $\begin{array}{c}{\left(6m+2\right)}^2=\left(1\right){\left(6m\right)}^2{\left(2\right)}^0+2{\left(6m\right)}^{2-1}\left(2\right)+1{\left(6m\right)}^{2-2}{\left(2\right)}^2\\ =36m^2+2\times 6m\times 2+4\\ =36m^2+24m+4\end{array}$

Thus, the expansion of the expression ${\left(6m+2\right)}^2$ is $36m^2+24m+4$ .