Problem 26 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using the FOIL method and simplify. $$(7 y-3)(2 y-1)$$

Step-by-Step Solution

Answer

The simplified product is \(14y^2 - 13y + 3\).

Step 1: Apply FOIL Method

The FOIL method involves multiplying two binomials by following these four steps: **F**irst, **O**uter, **I**nner, **L**ast. Let's break it down for the expression \((7y - 3)(2y - 1)\).**First:** Multiply the first terms of each binomial: \(7y \times 2y = 14y^2\). **Outer:** Multiply the outer terms of each binomial: \(7y \times -1 = -7y\). **Inner:** Multiply the inner terms of each binomial: \(-3 \times 2y = -6y\). **Last:** Multiply the last terms of each binomial: \(-3 \times -1 = 3\).

Step 2: Combine the Products

Now, add all the products from the FOIL steps together. The expression will be: \[14y^2 - 7y - 6y + 3\].

Step 3: Simplify the Expression

Combine like terms in the expression:Combine \(-7y\) and \(-6y\) to get \(-13y\). The simplified expression is:\[14y^2 - 13y + 3\].

Key Concepts

Binomials

In algebra, a binomial is an expression consisting of two terms. These terms are usually separated by a plus or minus sign. For example, in the expression

• \(7y - 3\)
• \(2y - 1\)

each expression is a binomial. Binomials are a type of polynomial, specifically ones with two terms. Understanding binomials is key, as they are foundational in algebraic calculations. The FOIL method is one of the most common techniques used to multiply two binomials, making it easier to handle such expressions. Whenever you see a binomial, think about how you might expand it, especially when dealing with multiple binomial expressions. Remember:

• Binomials have two terms.
• They are a specific instance of polynomials.
• Useful for performing operations like the FOIL method.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. They form the basis of much of algebra. These expressions don't just represent numbers or operations, but relationships and functional mappings. In the exercised problem

• \((7y - 3)(2y - 1)\)

the algebraic expression is formed by the multiplication of two binomials. It's important to remember:

• Expressions don't have equal signs like equations do.
• They can be simple, like single terms, or complex, involving operations on several terms.
• Main goal is to simplify and solve where possible.

Apply operations like FOIL, factorization, and simplification to manage and solve these expressions effectively.

Polynomials

Polynomials are expressions made up of terms, which include variables raised to whole-number exponents and coefficients. They are central to algebra, with various applications ranging from solving equations to modeling real-world situations. A binomial is a simple form of polynomial, only consisting of two terms, but polynomials can have many more terms. Here's what you should grasp:

• Terms in a polynomial can be constants or variables.
• A polynomial is usually characterized by its degree, which is the highest power of the variable in the expression.
• The structure allows for flexible manipulation and is fundamental to algebraic operations.

In the solution you encountered, the final expression \(14y^2 - 13y + 3\) is a polynomial that resulted from using the FOIL method on two binomials. Recognize the power and utility of polynomials in simplifying algebraic expressions and solving equations.