Problem 68 - Elementary Algebra

Question

Find the product. \((2 x-1)^{2}\).

Step-by-Step Solution

Answer

Answer: The product of the expression \((2x-1)^2\) is \(4x^2 - 4x + 1\).

Step 1: Write down the given expression in expanded form

First, write the expression \((2x-1)^2\) in its expanded form. This means multiplying the expression \((2x-1)\) by itself: \((2x-1)(2x-1)\).

Step 2: Apply the distributive property

Now, apply the distributive property, which states that for any real numbers a, b, and c, \((a+b)c = ac + bc\). In this case, we want to distribute \(2x-1\) through the second \((2x-1)\): \((2x - 1)(2x) – (2x - 1)(1)\).

Step 3: Distribute and simplify

Next, distribute the terms and simplify: \((2x - 1)(2x) = (2x)(2x) - (1)(2x)\) which simplifies to \(4x^2 - 2x\). \((2x - 1)(1) = (2x)(1)-(1)(1)\) which simplifies to \(2x - 1\).

Step 4: Combine the expressions

Now, we need to combine the expressions from steps 2 and 3: \(4x^2 - 2x - (2x - 1)\).

Step 5: Simplify the expression

Lastly, simplify the expression by removing the parentheses and combining like terms: \(4x^2 - 2x - 2x + 1\), which simplifies to \(4x^2 - 4x + 1\). So the product of \((2x-1)^2\) is \(4x^2 - 4x + 1\).

Key Concepts

Distributive Property

One of the foundational elements of algebra is the distributive property. This property is essential when you're multiplying a single term by a binomial or polynomial. The distributive property states that when you have a multiplication scenario such as \( a(b + c) \), you can distribute the multiplication of \( a \) to each term inside the parentheses, ending up with \( ab + ac \). This might seem abstract at first, so let's consider our given problem: \( (2x - 1)^2 \).

To apply this property, you'd multiply \( 2x \) by \( 2x \) and then \( 2x \) by \( -1 \) separately. Next, you'd do the same with \( -1 \)—multiply it by each term in the second binomial. It looks like this: \( (2x)(2x) + (2x)(-1) + (-1)(2x) + (-1)(-1) \). When working with the distributive property, keep attention to detail—especially with signs—as a slip can lead to incorrect results. After multiplying out the terms, proceed to the next crucial step: combining like terms.

Polynomial Expansion

While the distributive property helps you break down the multiplication of terms, polynomial expansion is the process of applying this property over more complex expressions, such as binomials and larger polynomials. For the case of binomial squares—like in our exercise where we have \( (2x - 1)^2 \)—we are essentially multiplying a binomial by itself, which requires polynomial expansion.

The formula to remember for the square of a binomial is \( (a - b)^2 = a^2 - 2ab + b^2 \). By using this pattern, you can quickly expand the binomial without having to distribute each term individually; however, understanding that this formula comes from the distributive property helps grasp why it works. Applying this directly to our exercise would look like \( (2x)^2 - 2(2x)(1) + (1)^2 \), ultimately simplifying to \( 4x^2 - 4x + 1 \) as a final expanded form of the binomial square.

Simplifying Expressions

Once we have expanded our expression using the distributive property and polynomial expansion, our next task is to simplify the expression. This simplification process combines like terms and rearranges the expression into its simplest form, making it easier to understand and evaluate. Like terms are those that have the exact same variable part, like \( 4x^2 \) and \( 3x^2 \) or \( 2x \) and \( 5x \), while terms like \( 4x^2 \) and \( 2x \) are not alike because they have different exponents.

In our problem, after expanding, we need to combine the terms that are similar, which are the \( -2x \) terms emerging from the distribution. So, we consolidate \( 4x^2 - 2x - 2x + 1 \) into \( 4x^2 - 4x + 1 \). Simplification clarifies the expression's structure, which can make further algebraic operations more straightforward. Always remember to check your work to ensure that no like terms are left uncombined and that the expression is as condensed as it can be.