Problem 21 - College Algebra

Question

Find each product. $$(x-5)(x+3)$$

Step-by-Step Solution

Answer

The product of the given binomials \(x-5\) and \(x+3\) is \(x^2 - 2x - 15\).

Step 1: Apply the FOIL Method

Use the FOIL method which stands for First, Outer, Inner, Last. Multiply the first terms in each binomial, then the outside terms, the inside terms, and finally the last terms of each binomial.

Step 2: Multiply the First Terms

Multiply the first terms in each binomial, i.e., \(x\) and \(x\), to get \(x^2\)

Step 3: Multiply the Outer Terms

Multiply the outer terms, i.e., \(x\) and 3 to get \(3x\)

Step 4: Multiply the Inner Terms

Multiply the inner terms, i.e. -5 and \(x\) to get \(-5x\)

Step 5: Multiply the Last Terms

Multiply the last terms, i.e. -5 and 3, to get \(-15\)

Step 6: Combine Like Terms

Combine the like terms \(3x\) and \(-5x\) to get \(-2x\). So, the expression becomes \(x^2-2x-15\)

Key Concepts

Understanding Binomials

A binomial is a type of polynomial that contains exactly two terms. These terms are usually combined by addition or subtraction. In the exercise, the binomials are

• \((x-5)\)
• \((x+3)\)

Both have a variable part, \(x\), and a constant part. The simplicity of binomials makes them perfect for practicing multiplication techniques like the FOIL method. When you multiply binomials, you expand them into a polynomial expression, which might contain more than two terms. Understanding how binomials work paves the way for mastering more complex polynomial operations.

Multiplying Polynomials with the FOIL Method

The FOIL method is a technique used specifically for multiplying binomials. It's an acronym that stands for First, Outer, Inner, Last. Here's how it works:

• First: Multiply the first terms in each binomial, \((x \cdot x = x^2)\).
• Outer: Multiply the outer terms, \((x \cdot 3 = 3x)\).
• Inner: Multiply the inner terms, \((-5 \cdot x = -5x)\).
• Last: Multiply the last terms, \((-5 \cdot 3 = -15)\).

This step-by-step approach ensures you correctly expand the binomials without missing any cross-product terms. Completing each step produces four terms that you'll later simplify by combining like terms.

Combining Like Terms

Once you have expanded the binomial using the FOIL method, you’ll need to simplify the expression by combining like terms. Like terms have the same variables raised to the same power. In the expanded expression \(x^2 + 3x - 5x - 15\), we look for terms with the variable \(x\).

• Combine \(3x\) and \(-5x\): These are like terms because they both have \(x\) as the variable. Adding them gets \(-2x\).
• Write the simplified expression: \(x^2 - 2x - 15\).

Combining like terms reduces the expression to its simplest form, making it easier to understand and use in further calculations. This step is crucial for fully simplifying polynomial results.