Chapter 8.4, Problem 120E

i.

To determine

To explain: How the number of terms related to exponent of a binomial.

Answer to Problem 120E

If $n$ is the exponent of a binomial, then there are $(n+1)$ terms in the binomial expansion.

Explanation of Solution

Given:

  ${(x+y)}^4=1x^4+4x^{3}y+6x^2{y}^2+4x{y}^3+1y^4$ .

Calculation:

By Binomial theorem,

  ${(x+y)}^n={\displaystyle \sum _{r=0}^n{}_n{C}_r{x}^{n-r}{y}^r}$ .

The total number of terms in the expansion is $(n+1)$ .

That is, if $n$ is the exponent of a binomial, then there are $(n+1)$ terms in the binomial expansion.

ii.

To determine

The number of terms in the given binomial expansion.

Answer to Problem 120E

The number of terms is $n+1$ .

Explanation of Solution

Given: The binomial ${(x+y)}^n$ .

Calculation:

Exponent of the given binomial is $n$ .

Therefore the number of terms is $n+1$ .