Answer :
To determine the correct binomial expansion, we can use the Binomial Theorem. The Binomial Theorem helps us expand expressions in the form of [tex]\((a + b)^n\)[/tex].
In this problem, it seems like we are looking at the expansion of [tex]\((x + 3)^3\)[/tex]. Let's expand it step-by-step:
1. Identify the expression to expand: [tex]\((x + 3)^3\)[/tex].
2. Apply the Binomial Theorem: According to the Binomial Theorem, [tex]\((a + b)^n\)[/tex] can be expanded as:
[tex]\[
\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
For [tex]\((x + 3)^3\)[/tex], [tex]\(a = x\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(n = 3\)[/tex].
3. Calculate each term:
- First term: [tex]\(\binom{3}{0} x^{3-0} 3^0 = 1 \cdot x^3 \cdot 1 = x^3\)[/tex]
- Second term: [tex]\(\binom{3}{1} x^{3-1} 3^1 = 3 \cdot x^2 \cdot 3 = 9x^2\)[/tex]
- Third term: [tex]\(\binom{3}{2} x^{3-2} 3^2 = 3 \cdot x \cdot 9 = 27x\)[/tex]
- Fourth term: [tex]\(\binom{3}{3} x^{3-3} 3^3 = 1 \cdot 1 \cdot 27 = 27\)[/tex]
4. Combine all terms to get the expansion:
[tex]\[
x^3 + 9x^2 + 27x + 27
\][/tex]
Given these calculations, we can see that option B, which is [tex]\(x^3 + 9x^2 + 27x + 27\)[/tex], matches the correct binomial expansion for [tex]\((x + 3)^3\)[/tex]. Therefore, the answer is option B.
In this problem, it seems like we are looking at the expansion of [tex]\((x + 3)^3\)[/tex]. Let's expand it step-by-step:
1. Identify the expression to expand: [tex]\((x + 3)^3\)[/tex].
2. Apply the Binomial Theorem: According to the Binomial Theorem, [tex]\((a + b)^n\)[/tex] can be expanded as:
[tex]\[
\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
For [tex]\((x + 3)^3\)[/tex], [tex]\(a = x\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(n = 3\)[/tex].
3. Calculate each term:
- First term: [tex]\(\binom{3}{0} x^{3-0} 3^0 = 1 \cdot x^3 \cdot 1 = x^3\)[/tex]
- Second term: [tex]\(\binom{3}{1} x^{3-1} 3^1 = 3 \cdot x^2 \cdot 3 = 9x^2\)[/tex]
- Third term: [tex]\(\binom{3}{2} x^{3-2} 3^2 = 3 \cdot x \cdot 9 = 27x\)[/tex]
- Fourth term: [tex]\(\binom{3}{3} x^{3-3} 3^3 = 1 \cdot 1 \cdot 27 = 27\)[/tex]
4. Combine all terms to get the expansion:
[tex]\[
x^3 + 9x^2 + 27x + 27
\][/tex]
Given these calculations, we can see that option B, which is [tex]\(x^3 + 9x^2 + 27x + 27\)[/tex], matches the correct binomial expansion for [tex]\((x + 3)^3\)[/tex]. Therefore, the answer is option B.
