Question
Which expression is the same as (2a + b)^3?
A. 8a^3 + B^3
B. 8a^3 + 4a^2 + 2ab^2+ b^3
• c. 8a^3+ 4a^2b+ 4ab^2+ b^3
D. 8a^3+12a^2b+6ab^2+b^3
A. 8a^3 + B^3
B. 8a^3 + 4a^2 + 2ab^2+ b^3
• c. 8a^3+ 4a^2b+ 4ab^2+ b^3
D. 8a^3+12a^2b+6ab^2+b^3
Answers
Answered by
GPT-4o mini
To find the expression equivalent to \((2a + b)^3\), we can use the binomial expansion formula, which states:
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\]
For \((2a + b)^3\), we identify \(x = 2a\), \(y = b\), and \(n = 3\).
The terms can be computed as follows:
\[
(2a + b)^3 = \sum_{k=0}^{3} \binom{3}{k} (2a)^{3-k} b^k
\]
Calculating each term:
1. For \(k = 0\):
\[
\binom{3}{0} (2a)^3 b^0 = 1 \cdot 8a^3 \cdot 1 = 8a^3
\]
2. For \(k = 1\):
\[
\binom{3}{1} (2a)^2 b^1 = 3 \cdot 4a^2 \cdot b = 12a^2b
\]
3. For \(k = 2\):
\[
\binom{3}{2} (2a)^1 b^2 = 3 \cdot 2a \cdot b^2 = 6ab^2
\]
4. For \(k = 3\):
\[
\binom{3}{3} (2a)^0 b^3 = 1 \cdot 1 \cdot b^3 = b^3
\]
Now, adding all these terms together, we have:
\[
(2a + b)^3 = 8a^3 + 12a^2b + 6ab^2 + b^3
\]
Thus, the correct expression that matches \((2a + b)^3\) is:
**D. \(8a^3 + 12a^2b + 6ab^2 + b^3\)**.
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\]
For \((2a + b)^3\), we identify \(x = 2a\), \(y = b\), and \(n = 3\).
The terms can be computed as follows:
\[
(2a + b)^3 = \sum_{k=0}^{3} \binom{3}{k} (2a)^{3-k} b^k
\]
Calculating each term:
1. For \(k = 0\):
\[
\binom{3}{0} (2a)^3 b^0 = 1 \cdot 8a^3 \cdot 1 = 8a^3
\]
2. For \(k = 1\):
\[
\binom{3}{1} (2a)^2 b^1 = 3 \cdot 4a^2 \cdot b = 12a^2b
\]
3. For \(k = 2\):
\[
\binom{3}{2} (2a)^1 b^2 = 3 \cdot 2a \cdot b^2 = 6ab^2
\]
4. For \(k = 3\):
\[
\binom{3}{3} (2a)^0 b^3 = 1 \cdot 1 \cdot b^3 = b^3
\]
Now, adding all these terms together, we have:
\[
(2a + b)^3 = 8a^3 + 12a^2b + 6ab^2 + b^3
\]
Thus, the correct expression that matches \((2a + b)^3\) is:
**D. \(8a^3 + 12a^2b + 6ab^2 + b^3\)**.
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