In this problem, we want to find the first three terms of this binomial expansion. There are several ways to approach this:
- attempt to multiply the terms
- use Pascal's Triangle
- use the Binomial Theorem
Since we only want the first three terms, we will use the binomial theorem, which states
[tex](a+b)^n=_nC_0a^nb^0+_nC_1a^{n-1}b^1+_nC_2a^{n-2}b^2+...+_nC_na^0b^n[/tex]
Where n represents the exponent of the binomial, and a and be represent the terms inside the parentheses. We use the combination function for the coefficients, n choose r objects.
From the given information, we know
[tex]\begin{gathered} n=9 \\ \\ a=x^2 \\ \\ b=6 \end{gathered}[/tex]
To find the first 3 terms, we will substitute the given information into the formula:
[tex]_9C_0(x^2)^96^0+_9C_1(x^2)^86^1+_9C_2(x^2)^76^2[/tex]
Simplifying the first term, we have
[tex]_9C_0(x^2)^96^0=1\cdot x^{18}\cdot1=x^{18}[/tex]
The second term:
[tex]_9C_1(x^2)^86^1=9\cdot x^{16}\cdot6=54x^{16}[/tex]
And the third term:
[tex]_9C_2(x^2)^76^2=36\cdot x^{14}\cdot36=1296x^{14}[/tex]
Putting each of those together, we get the first three terms:
[tex]\boxed{x^{18}+54x^{16}+1296x^{14}}[/tex]
