Problem 68 - Algebra 1: Concepts and Skills

Question

Find the product. \((2 x-4)(4 x-2)\)

Step-by-Step Solution

Answer

The product of \( (2 x-4)(4 x-2) \) is \( 8x^2 - 20x + 8 \)

Step 1: Apply the distributive (FOIL) property

To solve \( (2x - 4)(4x - 2) \), begin by distributing each term in the first parenthesis with each term in the second one using the FOIL method. It results to \( 2x * 4x - 2x * 2 - 4 * 4x + 4 * 2 \). This simplifies to \( 8x^2 - 4x - 16x + 8 \)

Step 2: Simplify

Combine like terms by adding the coefficients of \( -4x \) and \( -16x \) together to simplify the expression. The result is \( 8x^2 - 20x + 8 \)

Key Concepts

Distributive Property

The distributive property is an essential concept in algebra that allows us to simplify expressions and solve equations. This property involves distributing, or spreading out, multiplication over addition or subtraction. For any numbers or algebraic expressions, the distributive property can be represented as:

• \( a(b+c) = ab + ac \)

In the context of polynomials and algebraic expressions, the distributive property is often used in a method called FOIL. FOIL stands for First, Outer, Inner, and Last, referring to the terms in each binomial that need to be multiplied. Using the distributive property in this way ensures that we account for each pair of terms in two binomials, making sure nothing is omitted.

In our example, with \((2x - 4)(4x - 2)\), the distributive property helps us break down the problem into manageable parts, leading to the expression \(8x^2 - 4x - 16x + 8\). After applying the FOIL method, we further simplify it by combining like terms to eventually solve the equation.

Polynomials

Polynomials are a fundamental structure in algebra and involve expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A polynomial can have one or multiple terms, also called monomials. For instance, the expression \(8x^2 - 20x + 8\) from our exercise is a polynomial with three terms.

• The term \(8x^2\) is known as the quadratic term because of its exponent 2.
• The term \(-20x\) is the linear term, characterized by the exponent 1 on x.
• The constant term is the number 8, which has no variable attached.

Recognizing each of these components is important in understanding and manipulating polynomials. The degree of a polynomial is determined by the highest power of the variable in the expression. In our case, \(8x^2\), the highest degree is 2, making it a quadratic polynomial.
Understanding polynomials allows us to use operations like the distributive property more effectively and is essential in solving equations in algebra.

Algebraic Expressions

Algebraic expressions form the building blocks of algebra and consist of numbers, variables, and operations such as addition, subtraction, multiplication, and division. These expressions allow us to describe and solve a wide range of mathematical problems. In an expression like \(2x - 4\), for example, \(2x\) is a term consisting of a coefficient (2) and a variable (x), whereas -4 is a constant term.

When working with algebraic expressions, one of the key skills is to manipulate these expressions through simplification, expansion, and factoring. Each of these processes takes practice:

• **Simplification** involves combining like terms, as seen in the second step of our exercise with \(-4x - 16x\).
• **Expansion** often involves applying the distributive property, particularly for multiplying binomials, as demonstrated through the FOIL method.
• **Factoring** is essentially the reverse process of expansion, where expressions are rewritten as the product of simpler expressions.

Understanding algebraic expressions at a fundamental level prepares you for more advanced topics in algebra, equipping you with the tools needed for problem-solving.