Chapter 9.5, Problem 18E

To determine

To expand: The expression ${\left(x+1\right)}^4$ by using the Binomial Theorem.

Answer to Problem 18E

The expansion of the expression ${\left(x+1\right)}^4$ is $x^4+4x^3+6x^2+4x+1$ .

Explanation of Solution

Given information: The expression is ${\left(x+1\right)}^4$ .

Theorem used:Binomial Theorem: ${\left(x+y\right)}^n=x^n+n{x}^{n-1}y+\cdots +nx{y}^{n-1}+y^n$ .

Calculation:

Here, $n=4$ .

First write out the Pascal’s triangle such that the row that begins with 1, 4.

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

From the Pascal’s triangle, it is observed that the binomial coefficients of the expression are 1, 4, 6, 4 and 1.

Expand the expression ${\left(x+1\right)}^4$ by using formula in Binomial theorem as follows.

  $\begin{array}{c}{\left(x+1\right)}^4=\left(1\right)x^4+4x^{4-1}\left(1\right)+6x^{4-2}{\left(1\right)}^2+4x^{4-3}{\left(1\right)}^3+\left(1\right){\left(1\right)}^4\\ =x^4+4x^3+6x^2+4x+1\end{array}$

Thus, the expansion of the expression ${\left(x+1\right)}^4$ is $x^4+4x^3+6x^2+4x+1$ .