To determine the binomial expansion of [tex]\((m + 2)^4\)[/tex], let's carefully examine each part of the solution.
### Step-by-Step Solution:
1. Apply the Binomial Theorem:
The Binomial Theorem states that for any positive integer [tex]\(n\)[/tex]:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
Here, [tex]\(a = m\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(n = 4\)[/tex].
2. Calculate the Expansion:
The binomial expansion of [tex]\((m + 2)^4\)[/tex] will include the sum of terms derived from applying the binomial coefficients [tex]\(\binom{4}{k}\)[/tex] for [tex]\(k = 0\)[/tex] to [tex]\(4\)[/tex], combined with the corresponding powers of [tex]\(m\)[/tex] and [tex]\(2\)[/tex]:
[tex]\[
(m + 2)^4 = \binom{4}{0} m^4 2^0 + \binom{4}{1} m^3 2^1 + \binom{4}{2} m^2 2^2 + \binom{4}{3} m^1 2^3 + \binom{4}{4} m^0 2^4
\][/tex]
3. Simplify Each Term:
- [tex]\(\binom{4}{0} m^4 2^0 = 1 \cdot m^4 \cdot 1 = m^4\)[/tex]
- [tex]\(\binom{4}{1} m^3 2^1 = 4 \cdot m^3 \cdot 2 = 8m^3\)[/tex]
- [tex]\(\binom{4}{2} m^2 2^2 = 6 \cdot m^2 \cdot 4 = 24m^2\)[/tex]
- [tex]\(\binom{4}{3} m^1 2^3 = 4 \cdot m \cdot 8 = 32m\)[/tex]
- [tex]\(\binom{4}{4} m^0 2^4 = 1 \cdot 1 \cdot 16 = 16\)[/tex]
4. Combine the Terms:
[tex]\[
(m + 2)^4 = m^4 + 8m^3 + 24m^2 + 32m + 16
\][/tex]
Thus, the binomial expansion of [tex]\((m + 2)^4\)[/tex] is:
[tex]\[
m^4 + 8m^3 + 24m^2 + 32m + 16
\][/tex]
### Identification of Terms:
- Number of Terms:
Since this is a 4th power expansion, it has [tex]\(4 + 1 = 5\)[/tex] terms.
- Row of Pascal's Triangle:
The coefficients of the expansion are derived from the 5th row (0-indexed) of Pascal's Triangle:
[tex]\[
\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \binom{4}{4}
\][/tex]
### Conclusion:
- The binomial expansion of [tex]\((m + 2)^4\)[/tex] is:
[tex]\[
m^4 + 8m^3 + 24m^2 + 32m + 16
\][/tex]
- The expansion has 5 terms.
- The coefficients come from row 5 of Pascal's Triangle.
