Chapter 9.2, Problem 8E

Solution

To identify: The expansion of the binomial ${\left(p+q\right)}^9$ by using Pascal’s triangle.

The expanded form of the binomial is

  ${\left(p+q\right)}^9=p^9+9p^{8}q+36p^7{q}^2+84p^6{q}^3+126p^5{q}^4+126p^4{q}^5+84p^3{q}^6+36p^2{q}^7+9p{q}^8+q^9$

Given information:

The binomial is ${\left(p+q\right)}^9$ .

Explanation:

Consider the given binomial

The power of the given binomial is $9$ so use the ninth-row of pascal's triangle.

  

So, the element of the $9$ th row of the pascal's triangle are $\left(1,9,36,84,126,126,84,36,9,1\right)$ Which is the coefficient of the consecutive terms.

Binomial theorem states that:

  ${\left(a+b\right)}^n=P_0{\text{a}}^1+P_1{a}^{n-1}b+\mathrm{....}+P_n{\text{b}}^n\text{here}{P}_0,P_1,P_2,P_3,P_n$ are the numbers in the nth row of pascal’s triangle.

Replace the coefficient of the consecutive terms.

  $\begin{array}{c}{\left(p+q\right)}^9=1{\left(p\right)}^9+9{\left(p\right)}^8\left(q\right)+36{\left(p\right)}^7{\left(q\right)}^2+84{\left(p\right)}^6{\left(q\right)}^3+126{\left(p\right)}^5{\left(q\right)}^4+126{\left(p\right)}^4{\left(q\right)}^5\\ +84{\left(p\right)}^3{\left(q\right)}^6+36{\left(p\right)}^2{\left(q\right)}^7+9\left(p\right){\left(q\right)}^8+1{\left(q\right)}^9\\ {\left(p+q\right)}^9=p^9+9p^{8}q+36p^7{q}^2+84p^6{q}^3+126p^5{q}^4+126p^4{q}^5+84p^3{q}^6+36p^2{q}^7+9p{q}^8+q^9\end{array}$