Problem 34 - A Graphical Approach to College Algebra

Question

Find each product. $$(2 z-1)\left(-z^{2}+3 z-4\right)$$

Step-by-Step Solution

Answer

\(-2z^3 + 7z^2 - 11z + 4\).

Step 1: Distribute the First Term

Start by distributing the first term in the binomial \[(2z) \cdot (-z^2 + 3z - 4) = 2z(-z^2) + 2z(3z) + 2z(-4)\]This results in the expression \[-2z^3 + 6z^2 - 8z\].

Step 2: Distribute the Second Term

Continue by distributing the second term of the binomial: \[-1 \cdot (-z^2 + 3z - 4) = -1(-z^2) + (-1)(3z) + (-1)(-4)\]This results in the expression \[z^2 - 3z + 4\].

Step 3: Combine the Polynomials

Add the expressions obtained from the previous steps:\[-2z^3 + 6z^2 - 8z + z^2 - 3z + 4\].

Step 4: Simplify the Expression

Combine like terms to simplify:- Combine the \(z^2\) terms: \(6z^2 + z^2 = 7z^2\).- Combine the \(z\) terms: \(-8z - 3z = -11z\).The simplified expression is:\[-2z^3 + 7z^2 - 11z + 4\].

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra. It allows you to multiply a single term with an expression inside parentheses. For example,

• when you have an expression like \(a(b + c)\), you distribute \(a\) to both \(b\) and \(c\).
• This gives you \(ab + ac\).

In the context of polynomial multiplication, the Distributive Property is applied repeatedly to every term.

In our exercise, the polynomial \((2z - 1)(-z^2 + 3z - 4)\) requires distributing both terms in the first binomial across all terms in the second polynomial.

• First, distribute \(2z\) to each term \((-z^2, 3z, -4)\), resulting in \(-2z^3, 6z^2,\) and \(-8z\).
• Next, distribute \(-1\) to each term \((-z^2, 3z, -4)\), resulting in \(z^2, -3z,\) and \(4\).

This method ensures that each term in the first polynomial is multiplied by each term in the second.

Binomials

Binomials are expressions that contain exactly two terms. For example, \((2z - 1)\) is a binomial because it consists of two separate parts: \(2z\) and \(-1\).

• Each term in a binomial can be a constant, a variable, or a combination involving coefficients.
• When multiplying a binomial by another polynomial, each term in the binomial is distributed over the other polynomial's terms.

In our example, we focus on how each section of the binomial \(2z - 1\) interacts with the trinomial \(-z^2 + 3z - 4\).

Each piece is handled separately, effectively breaking down the problem into smaller, more manageable parts.This strategic breakdown into parts shows why understanding binomials is essential when learning how to multiply larger polynomials efficiently.

Combining Like Terms

Combining like terms is the process of simplifying expressions by merging terms that have the same variable to the same power. This is an important step after multiplying polynomials to make the expression more concise.

• "Like terms" are those that have identical variables raised to the same exponent, even if their coefficients are different.
• For example, in the polynomial expression \(6z^2 + z^2 - 3z - 2z\), "like terms" \(6z^2\) and \(z^2\) can be combined to become \(7z^2\).

When you combine like terms, you simply add or subtract the coefficients.

In our exercise, combing like terms involves reducing the polynomial from \(-2z^3 + 6z^2 - 8z + z^2 - 3z + 4\) to the simplified form \(-2z^3 + 7z^2 - 11z + 4\). This simplification is critical for clearly understanding and efficiently solving complex expressions by reducing them to their simplest form.