Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, find the product. \((4 x+2)(6 x-4)\)

Step-by-Step Solution

Answer

The product is \(24x^2 - 4x - 8\).

Step 1: Expand the Expression

To find the product of the binomials \((4x + 2)(6x - 4)\), we will use the distributive property (also known as the FOIL method for binomials), which stands for First, Outer, Inner, Last. Write out the multiplication: \(4x \cdot 6x + 4x \cdot (-4) + 2 \cdot 6x + 2 \cdot (-4)\).

Step 2: Calculate Each Term

Now, calculate each term separately:- First: \(4x \cdot 6x = 24x^2\)- Outer: \(4x \cdot (-4) = -16x\)- Inner: \(2 \cdot 6x = 12x\)- Last: \(2 \cdot (-4) = -8\)

Step 3: Combine Like Terms

Combine the terms from the expanded expression:\(24x^2 - 16x + 12x - 8\).Group the like terms:\\(24x^2 + (-16x + 12x) - 8\).

Step 4: Simplify the Expression

Simplify by combining the like terms:\(-16x + 12x = -4x\).Therefore, the expression simplifies to:\(24x^2 - 4x - 8\).

Key Concepts

Distributive Property

The distributive property is a key principle in algebra that is often used for the multiplication of polynomials. The core idea is to distribute or spread out the multiplication of one term over the addition or subtraction inside a set of parentheses. In very simple terms, if you have an expression such as \(a(b + c)\), the distributive property allows you to multiply \(a\) by both \(b\) and \(c\), resulting in \(ab + ac\).

This property is not only helpful in simplifying expressions but is also a fundamental concept in shifting from arithmetic to algebra. By utilizing the distributive property, complex algebraic expressions become manageable and can be simplified into easier expressions to handle.

In the given exercise, the distributive property is applied to break down the multiplication of the binomials \((4x + 2)(6x - 4)\) into separate parts, ensuring each term in the first binomial multiplies with every term in the second. This helps in systematically tackling polynomial multiplication, making each step straightforward.

Binomial Product

The term "binomial" refers to an algebraic expression containing exactly two terms. The product of two binomials happens often in algebra. When multiplying binomials, each term of the first binomial multiplies with every term of the second binomial.

Consider the product \((a + b)(c + d)\). To find this, you need to multiply each term in the first binomial by each term in the second binomial, creating \(ac + ad + bc + bd\). This ensures that all possible combinations of terms have been included in the product.

In mathematical practice, understanding how to handle binomial products is essential since many algebraic problems are structured in this way. The exercise given exemplifies this perfectly by requiring you to multiply two binomials \((4x + 2)\) and \((6x - 4)\), ensuring each multiplication is carried out between the terms.

FOIL Method

The FOIL method is a specialized technique for multiplying two binomials. It's essentially a specific application of the distributive property tailored for two-term expressions. The term FOIL stands for First, Outer, Inner, Last and describes the order in which terms from the binomials are multiplied.

Here's how it breaks down:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the innermost terms.
• Last: Multiply the last terms in each binomial.

This method brings a systematic approach to polynomial multiplication, ensuring all terms are combined correctly.
For example, with the expressions \((4x + 2)(6x - 4)\), the FOIL method helps organize the multiplications: - First: \(4x \cdot 6x = 24x^2\)- Outer: \(4x \cdot (-4) = -16x\)- Inner: \(2 \cdot 6x = 12x\)- Last: \(2 \cdot (-4) = -8\)
Once these are calculated, you simply combine the like terms to simplify the product, making the FOIL method a reliable strategy in handling binomial products.