Problem 60 - College Algebra with Modeling and Visualization

Question

Multiply the binomials. $$(z-2)(4 z+3)$$

Step-by-Step Solution

Answer

The product is \(4z^2 - 5z - 6\).

Step 1: Apply the Distributive Property

To multiply the binomials \((z-2)(4z+3)\), apply the distributive property by multiplying each term in the first binomial by each term in the second. Start with: \[ z imes (4z+3) \] and \[ -2 imes (4z+3) \]

Step 2: Expand the Binomials

Calculate the individual products for each term:- \(z \times 4z = 4z^2\)- \(z \times 3 = 3z\)- \(-2 \times 4z = -8z\)- \(-2 \times 3 = -6\)

Step 3: Combine Like Terms

Combine the like terms from the expanded expression:- The expression is \(4z^2 + 3z - 8z - 6\)- Combine the \(z\) terms: \(3z - 8z = -5z\).- Thus, the simplified result is: \(4z^2 - 5z - 6\).

Key Concepts

Distributive Property

The Distributive Property is a key mathematical principle that simplifies multiplication over addition or subtraction. When multiplying two binomials, this property comes to the rescue. In our example, \((z-2)(4z+3)\), we distribute each term in the first binomial to each term in the second binomial. First, distribute \(z\) over the second binomial:

• \(z \times 4z\)
• \(z \times 3\)

Next, do the same for \(-2\):

• \(-2 \times 4z\)
• \(-2 \times 3\)

Breaking down each multiplication step helps in organizing your work and ensuring no terms are skipped over, which makes simplifying equations much easier.

Combining Like Terms

Once you have used the Distributive Property to expand your binomial multiplication, your result will include several terms that can often look a bit messy. This is where combining like terms becomes essential. In our example, after distributing, we get:\(4z^2 + 3z - 8z - 6\)."Like terms" are terms that have the same variable raised to the same power. Hence, \(3z\) and \(-8z\) are like terms because they both contain \(z\). Combine these like terms by simply doing the arithmetic:

• \(3z - 8z = -5z\)

So, when you combine, it results in:\(4z^2 - 5z - 6\).Looking for and combining like terms reduces the number of terms in your expression, leading to a simpler and easier-to-understand answer.

Polynomial Expansion

Polynomial expansion is the result of applying the Distributive Property to multiply out binomials or polynomials. In our specific example, \((z-2)(4z+3)\), we achieve expansion through systematic distribution and multiplication. By the end of this process, the expanded form of our original expression is \(4z^2 - 5z - 6\).This expanded form is a quadratic polynomial since it includes \(z^2\), z, and constant terms. The expansion transforms an expression that appears compact into one that displays each term distinctly. Mastering this technique is essential for simplifying expressions and solving more complex algebraic equations that involve multiple polynomial terms.