Perform the indicated operations. See Examples 2–6.
$-z^3(9-z)+4z(2+3z)$

Verified step by step guidance

1. First, recognize that the expression involves distributing terms and combining like terms. The expression is $-z^3(9-z)+4z(2+3z)$.
2. Distribute $-z^3$ across the terms inside the first parentheses: multiply $-z^3$ by $9$ and then by $-z$ separately.
3. Distribute $4z$ across the terms inside the second parentheses: multiply $4z$ by $2$ and then by $3z$ separately.
4. After distribution, write out all the terms explicitly and combine like terms by adding or subtracting coefficients of the same powers of $z$.
5. Simplify the expression by combining the coefficients and writing the final polynomial in standard form, ordered by descending powers of $z$.

Key Concepts

Polynomial Operations

Polynomial operations involve addition, subtraction, multiplication, and division of polynomial expressions. Understanding how to combine like terms and apply distributive properties is essential for simplifying expressions like the given problem.

Exponent Rules

Exponent rules govern how to handle powers of variables, such as multiplying powers with the same base or raising a power to another power. Recognizing that z^3 means z multiplied by itself three times helps in correctly expanding and simplifying terms.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is crucial for expanding expressions like -z^3(9 - z) and 4z(2 + 3z), allowing each term inside the parentheses to be multiplied by the term outside.