Question

1. $(p + q)^9$ in exercises 5 - 8, expand the binomial using pascals triangle to find the coefficients.

Explanation:

Step1: Recall Pascal's triangle row for power 9

The 10th row (since we start counting rows of Pascal's triangle from 0) of Pascal's triangle gives the coefficients for \((a + b)^9\). The row is 1, 9, 36, 84, 126, 126, 84, 36, 9, 1.

Step2: Write out the binomial expansion formula

The binomial expansion of \((p + q)^n=\sum_{k = 0}^{n}\binom{n}{k}p^{n - k}q^{k}\). For \(n = 9\), we have \((p+q)^9=1p^{9}q^{0}+9p^{8}q^{1}+36p^{7}q^{2}+84p^{6}q^{3}+126p^{5}q^{4}+126p^{4}q^{5}+84p^{3}q^{6}+36p^{2}q^{7}+9p^{1}q^{8}+1p^{0}q^{9}\).

Step3: Simplify the terms

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

Answer:

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