Problem 68 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (3 x+2)(4 x-2) $$

Step-by-Step Solution

Answer

\(12x^2 + 2x - 4\)

Step 1: Apply FOIL method

\((3x+2)(4x-2)\):

• First: \(3x \cdot 4x = 12x^2\)
• Outer: \(3x \cdot (-2) = -6x\)
• Inner: \(2 \cdot 4x = 8x\)
• Last: \(2 \cdot (-2) = -4\)

Step 2: Combine like terms

\(12x^2 - 6x + 8x - 4 = 12x^2 + 2x - 4\)

Key Concepts

FOIL Method

The FOIL method is an essential technique in algebra often used to multiply two binomials. It stands for First, Outer, Inner, Last, which refers to the terms in the binomial expressions that need to be multiplied together. Let's break down how this works:

• First: Multiply the first terms of each binomial. For \((3x + 2)(4x - 2)\), it is \(3x \times 4x = 12x^2\).
• Outer: Multiply the outer terms of the binomials, which are \(3x \times -2 = -6x\).
• Inner: Multiply the inner terms, \(2 \times 4x = 8x\).
• Last: Multiply the last terms of each binomial: \(2 \times -2 = -4\).

Combine all these products: 12x² - 6x + 8x - 4.
Now, combine like terms, yielding the final expanded expression: 12x² + 2x - 4.
The FOIL method is a straightforward strategy that simplifies multiplying binomials effectively, making it a must-know for students.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients arranged in terms. These terms are comprised of variables raised to whole number exponents and constant coefficients.
When discussing polynomials, it's important to understand the following:

• Terms: Components of a polynomial separated by plus or minus signs. In \(3x + 2\), there are two terms: \(3x\) and \(2\).
• Degree: The highest exponent of the variable in the polynomial determines its degree. For example, in \(12x^2 + 2x - 4\), the degree is 2.
• Coefficients: Numerical factors that multiply the variable(s). In \(12x^2\), \(12\) is the coefficient.
• Like Terms: Terms with the same variable raised to the same power can be combined. In our exercise, \(-6x\) and \(8x\) are like terms.

Polynomials form the basis for more complex algebraic structures and are crucial for understanding algebraic expressions and equations.
They are foundational to algebra, enabling students to manipulate and solve equations efficiently.

Algebraic Expressions

An algebraic expression is a mathematical phrase that contains numbers, variables, and operators (such as +, -, *, /). These expressions do not include an equality sign, which distinguishes them from equations. Algebraic expressions are fundamental in forming more complex mathematical sentences. To understand them better, here are some key components:

• Constants: These are the fixed numbers in an expression, such as the \(2\) and \(-2\) in our original problem.
• Variables: Symbols (usually letters) that represent unknown numbers. In \(3x + 2\), \(x\) is a variable.
• Operators: Symbols representing operations that need to be performed. These include addition (+), subtraction (-), multiplication (*), and division (/).
• Terms: Parts of the expression separated by + or - signs, each consisting of a variable, number, or both. For example, \(3x\) is a term.

Algebraic expressions are essential for constructing and solving real-world problems in mathematics.
The ability to manipulate these expressions allows students to develop a stronger fluency in algebra.