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

Verified step by step guidance

1. Recognize that \((y + 2)^3\) is a binomial raised to the third power, which can be expanded using the Binomial Theorem.
2. The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = y\), \(b = 2\), and \(n = 3\).
3. Apply the Binomial Theorem: \((y + 2)^3 = \binom{3}{0} y^3 (2)^0 + \binom{3}{1} y^2 (2)^1 + \binom{3}{2} y^1 (2)^2 + \binom{3}{3} y^0 (2)^3\).
4. Calculate each binomial coefficient: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), \(\binom{3}{3} = 1\).
5. Substitute the coefficients and simplify each term: \(1 \cdot y^3 \cdot 1 + 3 \cdot y^2 \cdot 2 + 3 \cdot y \cdot 4 + 1 \cdot 1 \cdot 8\).

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ, where n is a non-negative integer. It states that (a + b)ⁿ can be expressed as the sum of terms involving binomial coefficients, which represent the number of ways to choose elements from a set. This theorem is essential for expanding polynomials like (y + 2)³.

Polynomial Expansion

Polynomial expansion involves rewriting a polynomial expression in a simplified form by multiplying out the factors. For example, expanding (y + 2)³ requires applying the Binomial Theorem or using the distributive property to combine like terms. Understanding how to expand polynomials is crucial for solving algebraic expressions and equations.

Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the expansion of a binomial expression, represented as C(n, k) or 'n choose k'. They indicate the number of ways to choose k elements from a set of n elements and are calculated using the formula n! / (k!(n-k)!). These coefficients play a key role in determining the coefficients of the terms in the expanded form of (y + 2)³.