Find each product. See Examples 5 and 6. (2z^4-3y)^2

Verified step by step guidance

1. Recognize that the expression $(2z^4-3y{)}^2$ is a binomial squared, which can be expanded using the formula $(a-b{)}^2=a^2-2ab+b^2$.
2. Identify $a=2z^4$ and $b=3y$ in the expression $(2z^4-3y{)}^2$.
3. Calculate $a^2$ by squaring $2z^4$, which results in $(2z^4{)}^2=4z^8$.
4. Calculate $2ab$ by multiplying $2$, $a=2z^4$, and $b=3y$, which results in $2\times 2z^4\times 3y=12z^{4}y$.
5. Calculate $b^2$ by squaring $3y$, which results in $(3y{)}^2=9y^2$.

Key Concepts

Polynomial Expansion

Polynomial expansion involves multiplying out expressions to express them in a standard form. In this case, the expression (2z^4 - 3y)^2 requires applying the distributive property or the binomial theorem to expand the square of a binomial. This process results in a polynomial that combines like terms.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem simplifies the process of expansion, especially for higher powers, by systematically calculating each term.

Combining Like Terms

Combining like terms is a fundamental algebraic process where terms with the same variable and exponent are added or subtracted. After expanding a polynomial, it is essential to identify and combine these terms to simplify the expression into its most concise form, ensuring clarity and ease of further calculations.