Question
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Tutor-Verified Answer
Answer
\((a + 2)^3 = a^3 + 6a^2 + 12a + 8\)
Solution
To expand \((a + 2)^3\), you can use the binomial theorem or simply multiply the expression out step by step.
**Using the Binomial Theorem:**
\[
(a + 2)^3 = a^3 + 3a^2(2) + 3a(2)^2 + (2)^3
\]
\[
= a^3 + 6a^2 + 12a + 8
\]
**Step-by-Step Multiplication:**
1. First, multiply \((a + 2)\) by itself:
\[
(a + 2)(a + 2) = a^2 + 4a + 4
\]
2. Then, multiply the result by \((a + 2)\):
\[
(a^2 + 4a + 4)(a + 2) = a^3 + 6a^2 + 12a + 8
\]
**Final Expanded Form:**
\[
(a + 2)^3 = a^3 + 6a^2 + 12a + 8
\]
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Simplify this solution Bonus Knowledge
To expand \( (a+2)^{3} \), you can use the binomial theorem, which states that \( (x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k} \). Here, \( x = a \), \( y = 2 \), and \( n = 3 \). The expansion is: \[ (a + 2)^{3} = \binom{3}{0} a^{3} (2)^{0} + \binom{3}{1} a^{2} (2)^{1} + \binom{3}{2} a^{1} (2)^{2} + \binom{3}{3} a^{0} (2)^{3} \] Calculating the coefficients: \[ = 1 \cdot a^{3} + 3 \cdot a^{2} \cdot 2 + 3 \cdot a \cdot 4 + 1 \cdot 8 \] \[ = a^{3} + 6a^{2} + 12a + 8 \] So, \( (a+2)^{3} = a^{3} + 6a^{2} + 12a + 8 \).
