Question

what is the binomial expansion of $(a + 2b)^2$ using pascal’s triangle?
a. $a^2 + 4ab + 2b^2$
b. $a^2 + 4ab + 4b^2$
c. $a^2 + 2ab + 2b^2$
d. $a^2 + 2ab + 4b^2$

Explanation:

Step1: Recall Pascal's Triangle for n=2

For a binomial \((x + y)^n\), the coefficients from Pascal's Triangle for \(n = 2\) are 1, 2, 1. The general form of binomial expansion is \((x + y)^2=1\cdot x^2+2\cdot x\cdot y + 1\cdot y^2\).

Step2: Substitute \(x = a\) and \(y=2b\)

Substitute \(x=a\) and \(y = 2b\) into the formula:
First term: \(1\cdot a^2=a^2\)
Second term: \(2\cdot a\cdot(2b)=4ab\)
Third term: \(1\cdot(2b)^2 = 4b^2\)
So \((a + 2b)^2=a^2+4ab + 4b^2\)

Answer:

b. \(a^{2}+4ab + 4b^{2}\)