Chapter 8.4, Problem 122E

(a)

To determine

The relation between the exponents of a binomial with the number of terms in theexpansion.

Answer to Problem 122E

In each binomial the number of terms is one more than its exponents.

Explanation of Solution

Given:

  $\begin{array}{l}\begin{array}{cc}{(x+y)}^4=1x^4+4x^{3}y& +6x^3{y}^2+4x{y}^3+1y^4\\ {(x+y)}^3=1x^2+5x^{4}y& +10x^3{y}^2+10x^2{y}^3+5x{y}^4+1y^2\\ & \end{array}\\ \begin{array}{cc}{(x+y)}^6=1x^2+6x^{2}y+15x^4{y}^2+20x^3{y}^3+15x^2{y}^4& +6x{y}^3+1y^2\\ & \end{array}\end{array}$

The number of terms in the expansion of the binomial is one more the exponent of the binomial and exponent of ${\left(x+y\right)}^3$ is 3.

And number of terms in this expansion is 4.

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

And number of terms in this expansion is 5.

The number of terms in the expansion of the binomial is one more the exponent of the binomial and exponent of ${\left(x+y\right)}^6$ is 6.

And number of terms in this expansion is 7.

From the given expansions,

It can be seen that in each binomial the number of terms is one more than its exponents.

(b)

To determine

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

Answer to Problem 122E

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

Explanation of Solution

Given:

  $\begin{array}{l}\begin{array}{cc}{(x+y)}^4=1x^4+4x^{3}y& +6x^3{y}^2+4x{y}^3+1y^4\\ {(x+y)}^3=1x^2+5x^{4}y& +10x^3{y}^2+10x^2{y}^3+5x{y}^4+1y^2\\ & \end{array}\\ \begin{array}{cc}{(x+y)}^6=1x^2+6x^{2}y+15x^4{y}^2+20x^3{y}^3+15x^2{y}^4& +6x{y}^3+1y^2\\ & \end{array}\end{array}$

From the observation of (a) the number of terms in the expansion of any binomial is 1 more than its exponent.

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