Expanding the Cubic Expression $(s + 2)^3$

The question asks us to expand the expression $(s + 2)^3$. This is a binomial expression raised to the power of 3. We can use the formula for the expansion of a cube of a binomial, which is $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

Applying the Binomial Expansion Formula

In our given expression $(s + 2)^3$, we can identify:

• $a = s$
• $b = 2$

Now, we substitute these values into the formula $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$:

<latex>(s + 2)^3 = (s)^3 + 3(s)^2(2) + 3(s)(2)^2 + (2)^3</latex>

Let's calculate each term step-by-step:

• The first term is <latex>(s)^3 = s^3</latex>.
• The second term is <latex>3(s)^2(2) = 3 \times s^2 \times 2 = 6s^2</latex>.
• The third term is <latex>3(s)(2)^2 = 3 \times s \times 4 = 12s</latex>.
• The fourth term is <latex>(2)^3 = 2 \times 2 \times 2 = 8</latex>.

Combining these terms, we get the expanded form:

<latex>(s + 2)^3 = s^3 + 6s^2 + 12s + 8</latex>

Comparing with Options

Now we compare our expanded expression with the given options:

• A. <latex>s^3 + 2s^2 + 12s + 8</latex>
• B. <latex>s^3 + 3s^2 + 6s + 8</latex>
• C. <latex>s^3 + 6s^2 + 12s + 8</latex>
• D. <latex>s^3 + 6s^2 + 6s + 8</latex>

Our result, <latex>s^3 + 6s^2 + 12s + 8</latex>, matches option C.

Therefore, the correct expansion of $(s + 2)^3$ is $s^3 + 6s^2 + 12s + 8$.

Revision Table: Key Algebraic Formulas

Understanding common algebraic expansion formulas is crucial for solving such problems.

Additional Information: Understanding Binomial Expansion

The expansion of <latex>(a+b)^n</latex> for any positive integer <latex>n</latex> can be found using the Binomial Theorem. The coefficients of the terms in the expansion <latex>(a+b)^n</latex> are given by the binomial coefficients <latex>\binom{n}{k}</latex> (read as "n choose k"), which can be found using Pascal's triangle or the formula <latex>\binom{n}{k} = \frac{n!}{k!(n-k)!}</latex>.

For <latex>n=3</latex>, the coefficients are <latex>\binom{3}{0}, \binom{3}{1}, \binom{3}{2}, \binom{3}{3}</latex>, which are 1, 3, 3, 1. These are the coefficients we saw in the expansion of <latex>(a+b)^3</latex>:

<latex>(a+b)^3 = \binom{3}{0}a^3b^0 + \binom{3}{1}a^2b^1 + \binom{3}{2}a^1b^2 + \binom{3}{3}a^0b^3</latex>

<latex>(a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3</latex>

<latex>(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3</latex>

This confirms the formula used to expand <latex>(s+2)^3</latex>.