Question

in exercises 5 - 8, expand the binomial using pascals triangle to find the coefficients. 5. $(x + y)^3$

Explanation:

Step1: Write out Pascal's triangle row for power 3

The 4th row (since we start counting rows from 0) of Pascal's triangle is 1, 3, 3, 1.

Step2: Apply binomial expansion formula

The binomial expansion of $(a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}$. For $(x + y)^3$, we have:
\[

$$\begin{align*} (x + y)^3&=1\times x^{3}y^{0}+3\times x^{2}y^{1}+3\times x^{1}y^{2}+1\times x^{0}y^{3}\\ &=x^{3}+3x^{2}y + 3xy^{2}+y^{3} \end{align*}$$

\]

Answer:

$x^{3}+3x^{2}y + 3xy^{2}+y^{3}$