Find each product. (y+2)³

Verified step by step guidance

1. Recognize that the expression \( (y+2)^3 \) represents a binomial raised to the third power, which means \( (y+2) \times (y+2) \times (y+2) \).
2. Use the binomial expansion formula for cubes: \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \). Here, \( a = y \) 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 combining all the terms: \( y^3 + 3y^2 \times 2 + 3y \times 4 + 8 \).
5. Simplify the coefficients in each term to get the final expanded expression.

Key Concepts

Binomial Expansion

Binomial expansion is the process of expanding expressions raised to a power, such as (a + b)^n, into a sum of terms involving coefficients, powers of a, and powers of b. It allows us to write the expression as a polynomial without directly multiplying multiple times.

Binomial Theorem and Coefficients

The Binomial Theorem provides a formula to expand (a + b)^n using binomial coefficients, which are found using combinations (n choose k). These coefficients determine the weight of each term in the expansion and can be found using Pascal's Triangle or the formula n!/(k!(n-k)!).

Polynomial Multiplication

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