Chapter 9.5, Problem 92E

(a)

To determine

Find the number of terms in the given expansion

Answer to Problem 92E

Number of terms in this expansion is 5

Explanation of Solution

Given:

  $\begin{array}{cc}{(x+y)}^4=1x^4+4x^{3}y& +6x^3{y}^2+4x{y}^3+1y^4\\ {(x+y)}^3=1x^2+5x^{4}y& +10x^3{y}^2+10x^2{y}^3+5x{y}^4+1y^2\\ & \end{array}$

  $\begin{array}{cc}(x+y{)}^{\prime }=1x^2+6x^{2}y+15x^4{y}^2+20x^3{y}^3+15x^2{y}^4& +6x{y}^3+1y^2\\ & \end{array}$

The number of terms in the expansion of the binomial is one more the exponent of the binomial is exponent of ${\left(x+y\right)}^4$ is 4

Number of terms in this expansion is 5

(b)

To determine

Find the number of terms in the expansion of ${\left(x+y\right)}^n$

Answer to Problem 92E

The number of terms in the expansion of ${\left(x+y\right)}^n$ is $n+1$

Explanation of Solution

Given:

  $\begin{array}{cc}{(x+y)}^4=1x^4+4x^{3}y& +6x^3{y}^2+4x{y}^3+1y^4\\ {(x+y)}^3=1x^2+5x^{4}y& +10x^3{y}^2+10x^2{y}^3+5x{y}^4+1y^2\\ & \end{array}$

  $\begin{array}{cc}(x+y{)}^{\prime }=1x^2+6x^{2}y+15x^4{y}^2+20x^3{y}^3+15x^2{y}^4& +6x{y}^3+1y^2\\ & \end{array}$

From the observation of (a) the number of terms in the expansion of ${\left(x+y\right)}^n$ is $n+1$