Find each product. See Examples 5 and 6. (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 $(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k$. Here, $a=y$, $b=2$, and $n=3$.
3. Calculate each term of the expansion: $(\genfrac{}{}{0ex}{}{3}{0})y^{3-0}{2}^0$, $(\genfrac{}{}{0ex}{}{3}{1})y^{3-1}{2}^1$, $(\genfrac{}{}{0ex}{}{3}{2})y^{3-2}{2}^2$, and $(\genfrac{}{}{0ex}{}{3}{3})y^{3-3}{2}^3$.
4. Evaluate the binomial coefficients: $(\genfrac{}{}{0ex}{}{3}{0})=1$, $(\genfrac{}{}{0ex}{}{3}{1})=3$, $(\genfrac{}{}{0ex}{}{3}{2})=3$, $(\genfrac{}{}{0ex}{}{3}{3})=1$.
5. Combine the terms to form the expanded expression: $1\cdot y^3+3\cdot y^2\cdot 2+3\cdot y\cdot 4+1\cdot 8$.

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 efficiently without multiplying the binomial repeatedly.

Polynomial Expansion

Polynomial expansion involves rewriting a polynomial expression in a simplified form, typically as a sum of terms. In the case of (y + 2)^3, expansion will yield a cubic polynomial. Understanding how to expand polynomials is crucial for simplifying expressions and solving equations in algebra.

Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the expansion of a binomial expression. They are denoted as C(n, k) or 'n choose k', representing the number of ways to choose k elements from a set of n elements. These coefficients are calculated using factorials and play a key role in determining the coefficients of each term in the expanded form of a binomial expression.