Problem 39 - Elementary Algebra

Question

Find the product. \((2 x-4)^{2}\).

Step-by-Step Solution

Answer

Answer: The product is 4x^2 -16x +16.

Step 1: Set up the expression for the product

Since the problem involves the square of a binomial, we rewrite the expression \((2x-4)^2\) as \((2x-4)(2x-4)\).

Step 2: Distribute the terms of the first binomial

Start by multiplying each term in the first binomial \((2x-4)\) with terms from the second binomial \((2x-4)\): \((2x-4)(2x-4) \\ = 2x(2x) + 2x(-4) -4(2x) -4(-4)\)

Step 3: Multiply the polynomials

Now, you'll multiply the polynomials from step 2: \(= 4x^2 -8x -8x +16\)

Step 4: Simplify the polynomial

Combine the like terms: \(= 4x^2 -16x +16\) So, the product of \((2x-4)^2\) is \(4x^2 -16x +16\).

Key Concepts

Binomials

A binomial is an algebraic expression that contains exactly two distinct terms. These terms are typically separated by either a plus (+) or a minus (-) sign. For example, in the expression \((2x-4)\), the terms are \(2x\) and \(-4\). Binomials are a special case of polynomials, which can contain two or more terms. Understanding binomials is a fundamental step in algebra because they form the basis for many polynomial operations, such as addition, subtraction, and multiplication.
When working with binomials, it's important to grasp the concept of combining these terms in various operations. Here are a few key points:

• Each term in a binomial may have coefficients and variables. Coefficients are the numerical factors, like the 2 in \(2x\).
• Variables represent unknown values and are often denoted by letters, such as \(x\) in \(2x\).
• The process of squaring a binomial involves multiplying the binomial by itself, which leads us to the next topic: multiplying polynomials.

Polynomial Multiplication

Polynomial multiplication, particularly involving binomials, is a process that requires the distribution of each term in one polynomial to every term in the other. In our exercise, we see this concept applied with \((2x-4)^2\). This square can be expanded as \((2x-4)(2x-4)\). This expansion showcases how each term in the first binomial is multiplied by each term in the second.
Here's a breakdown of the multiplication steps:

• Distribute the first term of the first binomial \(2x\) across both terms of the second binomial: \(2x \times 2x\) and \(2x \times -4\).
• Do the same for the second term \(-4\) in the first binomial: \(-4 \times 2x\) and \(-4 \times -4\).

When you multiply these out, you get each part of the expanded equation:

• \(2x \times 2x = 4x^2\)
• \(2x \times -4 = -8x\)
• \(-4 \times 2x = -8x\)
• \(-4 \times -4 = 16\)

This expansion and distribution process is necessary for tackling any polynomial multiplication, not just with binomials. Once you master it, you'll be able to solve and simplify a wide variety of algebraic expressions.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that involves combining like terms to reduce an expression to its simplest form. In our exercise, after multiplying the binomials, we arrive at the expression \(4x^2 -8x -8x +16\). This step is pivotal as it helps in reducing complicated expressions to more manageable ones.
The strategy for simplifying involves looking for terms with the same variables raised to the same power, known as 'like terms'. Here's how you can simplify the expression step-by-step:

• Identify like terms: In \(-8x\) and \(-8x\), both terms contain \(x\) as the variable and have the same exponent.
• Combine like terms: Add these like terms together to simplify, resulting in \(-16x\).
• Rewrite the expression: Combine all terms, \(4x^2 -16x +16\), which is the simplest form of the original expanded expression.

Simplifying expressions helps in making both understanding and further mathematical operations more efficient. It allows for clearer insights into the relationships between the different parts of an expression, making it a fundamental concept in algebra.