Chapter 9.5, Problem 19E

To determine

To expand: The expression ${\left(y-3\right)}^3$ by using the Binomial Theorem.

Answer to Problem 19E

The expansion of the expression ${\left(y-3\right)}^3$ is $y^3-9y^2+27y-27$ .

Explanation of Solution

Given information: The expression is ${\left(y-3\right)}^3$ .

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, $n=3$ .

First write out the Pascal’s triangle such that the row that begins with 1, 3.

  $\begin{array}{ccccccc}& & & 1& & & \\ & & 1& & 1& & \\ & 1& & 2& & 1& \\ 1& & 3& & 3& & 1\end{array}$

From the Pascal’s triangle, it is observed that the binomial coefficients of the expression are 1, 3, 3 and 1.

Expand the expression ${\left(y-3\right)}^3$ by using formula in Binomial theorem as follows.

  $\begin{array}{c}{\left(y-3\right)}^3=\left(1\right)y^3+3y^{3-1}\left(-3\right)+3y^{3-2}{\left(-3\right)}^2+1{\left(-3\right)}^3\\ =y^3-9y^2+27y-27\end{array}$

Thus, the expansion of the expression ${\left(y-3\right)}^3$ is $y^3-9y^2+27y-27$ .