Certainly! Let's find the first two non-zero terms in the binomial expansion of the expression [tex]\(\left[(2 + 3x)^6 + (2 - 3x)^6 \right]^2\)[/tex].
1. Express the Binomials: First, let's consider the expressions [tex]\((2 + 3x)^6\)[/tex] and [tex]\((2 - 3x)^6\)[/tex].
2. Combine the Binomials: Now, we combine these two binomials:
[tex]\[
(2 + 3x)^6 + (2 - 3x)^6
\][/tex]
3. Square the Sum: This becomes:
[tex]\[
\left[(2 + 3x)^6 + (2 - 3x)^6\right]^2
\][/tex]
4. Expand Each Binomial Individually: Instead of calculating each term directly (which would be very complex), we rely on knowing the expanded form:
[tex]\[
2125764x^{12} + 28343520x^{10} + 107075520x^8 + 84354048x^6 + 21150720x^4 + 1105920x^2 + 16384
\][/tex]
5. Identify the First Two Non-Zero Terms: From the expanded polynomial, the first two non-zero terms are:
[tex]\[
2125764x^{12} \quad \text{and} \quad 28343520x^{10}
\][/tex]
These represent the highest degree terms with non-zero coefficients in the expanded form.
Therefore, the first two non-zero terms in the binomial expansion of [tex]\(\left[(2 + 3x)^6 + (2 - 3x)^6 \right]^2\)[/tex] are:
[tex]\[
2125764x^{12} \quad \text{and} \quad 28343520x^{10}
\][/tex]
