Sure! Let's expand [tex]\((x + 3)^7\)[/tex] using the binomial theorem, which gives a systematic way to expand expressions of the form [tex]\((a + b)^n\)[/tex].
The binomial theorem states that each term in the expansion of [tex]\((x + 3)^7\)[/tex] will consist of a binomial coefficient, a power of [tex]\(x\)[/tex], and a power of [tex]\(3\)[/tex]. Here's the detailed step-by-step process:
1. Identify the components: The expression [tex]\((x + 3)^7\)[/tex] has two terms, [tex]\(x\)[/tex] and [tex]\(3\)[/tex]. We need to determine the contribution of each term.
2. Determine the binomial coefficients: For each term in the expansion, binomial coefficients can be found by evaluating combinations. In general, the k-th term has a coefficient given by the combination [tex]\( \binom{n}{k} \)[/tex], where [tex]\( n \)[/tex] is the power (in our case, 7) and [tex]\( k \)[/tex] ranges from 0 to 7.
3. Set up the terms: Each term in the series is made up of:
- The binomial coefficient [tex]\( \binom{7}{k} \)[/tex]
- The power of [tex]\(x\)[/tex] which decreases from 7 to 0
- The power of [tex]\(3\)[/tex] which increases from 0 to 7
4. Construct each term: So the expansion will be the sum of these individual terms.
Here’s how each term is constructed for [tex]\((x + 3)^7\)[/tex]:
- The first term is derived when [tex]\( k = 0 \)[/tex]: [tex]\( \binom{7}{0} x^7 (3^0) = 1 \cdot x^7 \cdot 1 = x^7 \)[/tex].
- The second term is derived when [tex]\( k = 1 \)[/tex]: [tex]\( \binom{7}{1} x^6 (3^1) \)[/tex].
- The third term is when [tex]\( k = 2 \)[/tex]: [tex]\( \binom{7}{2} x^5 (3^2) \)[/tex].
- Continue this pattern up to [tex]\( k = 7 \)[/tex]: [tex]\( \binom{7}{7} x^0 (3^7) \)[/tex].
5. Sum the terms: Add all these terms together to get the final expansion.
Thus, the expanded form of [tex]\((x + 3)^7\)[/tex] will be:
[tex]\[ x^7 + \binom{7}{1} x^6 (3) + \binom{7}{2} x^5 (3^2) + \cdots + \binom{7}{7} (3^7) \][/tex]
Each term involves a combination coefficient, a power of [tex]\(x\)[/tex], and a power of 3. These terms collectively give you the entire expansion of the binomial expression.
