Problem 85 - Algebra A Combined Function

Question

Multiply. See Section 5.6. \((2 x-1)^{2}\)

Step-by-Step Solution

Answer

The result of squaring \((2x-1)\) is \(4x^2 - 4x + 1\).

Step 1: Identify the Expression

We need to multiply the expression \((2x-1)^2\). This means we are squaring the binomial \((2x-1)\), or \((2x-1)\) multiplied by \((2x-1)\).

Step 2: Apply the Binomial Square Formula

Use the formula for squaring a binomial: \((a-b)^2 = a^2 - 2ab + b^2\). Here, set \(a = 2x\) and \(b = 1\).

Step 3: Square the First Term

Calculate \((2x)^2\). This results in \(4x^2\).

Step 4: Calculate Twice the Product of the Terms

Find \(-2ab\), which is \(-2(2x)(1) = -4x\).

Step 5: Square the Second Term

Calculate \((1)^2\), which is \(1\).

Step 6: Combine the Results

Combine all the terms from the binomial expansion: \(4x^2 - 4x + 1\).

Key Concepts

Squaring a Binomial

Squaring a binomial is a useful technique often employed in algebra for expanding expressions written in the form \((a - b)^2\) or \((a + b)^2\). This process involves the multiplication of a binomial by itself: in simpler terms, multiplying two identical binomial expressions.

The magical part about squaring a binomial is the use of a specific formula that eliminates the need for a lengthy multiplication process. The formula for squaring a binomial \((a-b)^2\) is represented by:

• \(a^2 - 2ab + b^2\)

These components are straightforward:

• \(a^2\) indicates the square of the first term.
• \(-2ab\) represents twice the product of the first and second terms.
• \(b^2\) involves squaring the second term.

So when you square a binomial, you're essentially calculating these three parts and adding them together. It's a handy shortcut that speeds up simplifying algebraic expressions.

Polynomial Multiplication

When dealing with polynomials, multiplication is a frequent operation and understanding it is crucial. Polynomial multiplication can take various forms, but the principle remains consistent: multiplying each term in one polynomial by every term in another.

In the context of squaring the binomial \((2x - 1)^2\), we perform the multiplication of \((2x - 1)\) by itself. The process involves:

• Multiplying each term of the first binomial by each term of the second binomial.
• Using techniques like the distributive property to ensure all terms are correctly multiplied and simplified.

This yields a new polynomial composed of terms from both initial binomials. When using formulas like the binomial square formula, it makes this multiplication more efficient by predicting the result straight away without needing to compute each pair of terms separately.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers, variables, and operators (like addition and subtraction) structured in a meaningful way. Working with algebraic expressions is crucial for solving various mathematical problems.

In our exercise, the expression \((2x - 1)^2\) is an example of an algebraic expression involving a variable \(x\). Let's break down the important aspects of algebraic expressions:

• Variables: Symbols that represent unknown numbers, commonly denoted as \(x, y, z,\) etc.
• Coefficients: Numbers appearing before variables, indicating how many times the variable is multiplied.
• Constants: Numbers on their own without accompanying variables, such as the \(-1\) in our example.
• Operators: Signs like "+" or "-" that denote operations between terms.

Understanding these components is critical for manipulating and simplifying expressions in algebra through processes like polynomial multiplication and binomial expansion. It’s all about combining terms and simplifying them to achieve a concise representation of the expression.