Problem 28 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using the FOIL method and simplify. $$(4 x-5 y)(3 x-y)$$

Step-by-Step Solution

Answer

The product is \(12x^2 - 19xy + 5y^2\).

Step 1: Understand the FOIL Method

The FOIL method is a technique for multiplying two binomials. FOIL is an acronym that stands for First, Outer, Inner, Last. Each term in the first binomial is multiplied by each term in the second binomial. We'll apply this to the given expressions.

Step 2: Multiply the First Terms

Multiply the first terms from each binomial: \((4x)(3x) = 12x^2\).

Step 3: Multiply the Outer Terms

Multiply the outer terms of the binomials: \((4x)(-y) = -4xy\).

Step 4: Multiply the Inner Terms

Multiply the inner terms of the binomials: \((-5y)(3x) = -15xy\).

Step 5: Multiply the Last Terms

Multiply the last terms of each binomial: \((-5y)(-y) = 5y^2\).

Step 6: Combine Like Terms

Add all the products from the FOIL method together and combine the like terms: \(12x^2 - 4xy - 15xy + 5y^2\).Combine terms with \(xy\): \(12x^2 - 19xy + 5y^2\).

Step 7: Write the Final Simplified Expression

The simplified product of the binomials \((4x-5y)(3x-y)\) is \(12x^2 - 19xy + 5y^2\).

Key Concepts

Multiplying Binomials

To multiply binomials, we often use an approach known as the FOIL method. This method is especially handy when dealing with two-term expressions, or binomials. Imagine you are multiplying a pair of binomials, such as given in the exercise: \((4x - 5y)(3x - y)\). Here's how FOIL bursts into action:

• First: Multiply the first terms of each binomial, resulting in \(4x \cdot 3x = 12x^2\).
• Outer: Multiply the outer terms, or those on the far ends: \(4x \cdot (-y) = -4xy\).
• Inner: Multiply the inner terms: \(-5y \cdot 3x = -15xy\).
• Last: Multiply the last terms: \(-5y \cdot (-y) = 5y^2\).

Each multiplication results in a term, and together these terms compose a polynomial expression. It’s important to follow this sequence to ensure every term is covered.

Algebraic Expressions

An algebraic expression can consist of constants, variables, and arithmetic operations. In the case of binomials like \(4x - 5y\) and \(3x - y\), they are composed of two terms each. When working with algebraic expressions, it's critical to recognize different components such as coefficients, variables, and operators.

• The coefficient is the numerical part of a term (e.g., 4 in \(4x\)).
• The variables are the symbols representing unknown values, like \(x\) and \(y\).
• The operators, such as addition or subtraction, dictate how terms interact with one another.

Knowing how to identify and manipulate these components in the expressions is essential when employing the FOIL method, as it helps to accurately apply multiplications and simplify results.

Combining Like Terms

Once you’ve multiplied out the terms using FOIL, you’ll have a new expression with multiple terms, like \(12x^2 - 4xy - 15xy + 5y^2\). Simplifying this requires combining like terms. "Like terms" are terms that have identical variable parts. For example, \(-4xy\) and \(-15xy\) are like terms because they both contain the variable part \(xy\). To combine them, you simply add or subtract their coefficients: \(-4 - 15 = -19\). So, combined they become \(-19xy\). Combining like terms is necessary for simplifying the expression into its most condensed form, which, in this case, leads us to \(12x^2 - 19xy + 5y^2\). By doing so, the expression becomes much easier to work with or interpret for further mathematical operations or analyses.