Find each product. See Example 5. (y + 2)³

Verified step by step guidance

1. Recognize that the expression \((y + 2)^3\) is a binomial raised to the third power, which means you need to expand it using the binomial cube formula.
2. Recall the binomial cube formula: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Here, identify \(a = y\) and \(b = 2\).
3. Calculate each term separately: \(a^3 = y^3\), \(3a^2b = 3 \times y^2 \times 2\), \(3ab^2 = 3 \times y \times 2^2\), and \(b^3 = 2^3\).
4. Write the expanded form by substituting the calculated terms: \(y^3 + 3y^2 \times 2 + 3y \times 4 + 8\).
5. Simplify the coefficients in each term to get the final expanded expression in terms of \(y\).

Key Concepts

Binomial Expansion

Binomial expansion is the process of expanding expressions raised to a power, such as (a + b)³. It uses the Binomial Theorem or Pascal's Triangle to find each term in the expanded form without direct multiplication.

Binomial Theorem

The Binomial Theorem provides a formula to expand powers of binomials: (a + b)^n = Σ [n choose k] a^(n-k) b^k. It helps calculate coefficients and powers systematically for each term in the expansion.

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by every term in another. For powers like (y + 2)³, this means multiplying (y + 2) by itself three times and combining like terms.