Find each product. See Examples 3–5. (2z-1)(-z^2+3z-4)

Verified step by step guidance

1. Distribute each term in the first polynomial \((2z - 1)\) to each term in the second polynomial \((-z^2 + 3z - 4)\).
2. Multiply \(2z\) by each term in \((-z^2 + 3z - 4)\): \(2z \cdot (-z^2)\), \(2z \cdot 3z\), and \(2z \cdot (-4)\).
3. Multiply \(-1\) by each term in \((-z^2 + 3z - 4)\): \(-1 \cdot (-z^2)\), \(-1 \cdot 3z\), and \(-1 \cdot (-4)\).
4. Combine all the products from the previous steps: \(2z \cdot (-z^2) + 2z \cdot 3z + 2z \cdot (-4) + (-1) \cdot (-z^2) + (-1) \cdot 3z + (-1) \cdot (-4)\).
5. Simplify the expression by combining like terms to get the final polynomial expression.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This process, often referred to as the distributive property, requires careful attention to combine like terms and ensure all products are accounted for. For example, in multiplying (2z - 1) by (-z^2 + 3z - 4), each term in the first polynomial must be multiplied by each term in the second.

Combining Like Terms

Combining like terms is a fundamental algebraic process where terms with the same variable and exponent are added or subtracted. This simplification is crucial after performing operations like multiplication, as it helps to express the polynomial in its simplest form. For instance, after multiplying the polynomials, you may end up with several terms that can be combined to reduce the expression.

Standard Form of a Polynomial

The standard form of a polynomial is a way of writing the polynomial where the terms are arranged in descending order of their degree. This format makes it easier to analyze the polynomial's properties, such as its degree and leading coefficient. After finding the product of (2z - 1)(-z^2 + 3z - 4), it is important to rearrange the resulting expression into standard form for clarity and further analysis.