Find each product. [(4y-1)+z]²

Verified step by step guidance

1. Recognize that the expression is a square of a binomial: \([(4y - 1) + z]^2\). This can be treated as \((A + B)^2\) where \(A = (4y - 1)\) and \(B = z\).
2. Recall the formula for the square of a binomial: \((A + B)^2 = A^2 + 2AB + B^2\).
3. Calculate \(A^2\) by squaring \((4y - 1)\): \((4y - 1)^2 = (4y)^2 - 2 \times 4y \times 1 + 1^2\).
4. Calculate \$2AB$ by multiplying \(2 \times (4y - 1) \times z\).
5. Calculate \(B^2\) by squaring \(z\): \(z^2\). Then, combine all parts to write the expanded expression: \(A^2 + 2AB + B^2\).

Key Concepts

Binomial Expansion

Binomial expansion involves expressing the square of a sum or difference of two terms as a trinomial. For example, (a + b)^2 expands to a^2 + 2ab + b^2. This formula helps simplify expressions like [(4y - 1) + z]^2 by treating (4y - 1) as one term and z as the other.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and exponentiation of polynomial expressions. Understanding how to multiply and combine like terms is essential when expanding expressions such as [(4y - 1) + z]^2 to simplify the result correctly.

Like Terms and Simplification

Like terms are terms in a polynomial that have the same variables raised to the same powers. After expanding an expression, combining like terms simplifies the expression into its simplest form, making it easier to interpret and use in further calculations.