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To find the volume of a cube, we use a simple formula. The volume (V) of a cube with side length (s) is calculated by cubing the side length. Mathematically, this is represented as: \[ V = s^3 \] This formula tells us that if we know the length of one side of the cube, we simply multiply that length by itself three times to get the volume. For instance, if each side of the cube is 3 cm, the volume is: \( V = 3^3 = 27 \text{ cm}^3 \) It’s a straightforward and essential concept in geometry. When dealing with algebraic expressions like \((x + 2)\), we apply the same principle, just with a bit more algebra involved.

The binomial theorem is a way to expand expressions that are raised to a power, such as \((x + 2)^3\). It provides a formula that simplifies the process of expansion. The binomial theorem for any expression \((a + b)^n\) is given by: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] Here, \(\binom{n}{k}\) represents the binomial coefficient, which is calculated as \(\frac{n!}{k!(n-k)!}\). Applying this to \((x + 2)^3\), where \(a = x\), \(b = 2\), and \(n = 3\), we get: \( (x + 2)^3 = x^3 + 3x^2(2) + 3x(2^2) + 2^3 \) After calculating, this expands to: \( x^3 + 6x^2 + 12x + 8 \). This theorem is extremely useful for simplifying polynomial expressions raised to a power.

Expanding cubic expressions involves transforming an expression like \((x + 2)^3\) into a polynomial. Here’s a step-by-step explanation of how it’s done using the binomial theorem: 1. Identify each term: In \((x + 2)^3\), we have \(a = x\) and \(b = 2\).
2. Apply the binomial formula: \( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \).
3. Substitute the values: Replace \(a\) and \(b\) with \(x\) and \(2\) respectively, giving us \((x + 2)^3 = x^3 + 3x^2(2) + 3x(2^2) + 2^3\).
4. Simplify each term: Calculate the terms individually: \(x^3\), \(6x^2\), \(12x\), and \(8\).
5. Combine the terms: Put all calculated terms together: \( x^3 + 6x^2 + 12x + 8 \).
By following these steps, you can expand any cubic expression accurately.