Problem 45 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x-3)^{2} $$

Step-by-Step Solution

Answer

The product of the binomial \( (x-3)^{2} \) is \( x^{2} - 6x + 9 \).

Step 1: Identify the binomial

Recognize the binomial to be squared as \( (x-3) \). Here, \( a = x \) and \( b = -3 \).

Step 2: Use binomial square formula

Apply the formula \( a^{2} + 2ab + b^{2} \), which gives \( x^{2} + 2 * x * -3 + (-3)^{2} \).

Step 3: Simplify the expression

Simplify the expression to obtain \( x^{2} - 6x + 9 \).

Key Concepts

Squaring a Binomial

Understanding how to square a binomial is a fundamental skill in algebra. Squaring a binomial essentially means multiplying the binomial by itself. A binomial is an algebraic expression that contains two terms, such as \(a + b\) or \(x - 3\), separated by a plus or minus sign.

To square a binomial, we use the binomial square formula, which is expressed as \( (a + b)^2 = a^2 + 2ab + b^2 \). This formula can be easily remembered as the square of the first term, plus twice the product of the two terms, plus the square of the second term.

For example, if you're given the exercise to find the product of \( (x-3)^2 \), you recognize your binomial \( (x-3) \) where \( a = x \) and \( b = -3 \). Applying the binomial square formula leads to \( x^2 - 6x + 9 \), which is the simplified final result.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In the context of our exercise, \( (x-3)^2 \), the expression represents a function of \(x\), involving a binomial square. Such expressions are foundational in algebra because they demonstrate how variables can interact and represent quantities in a generalized form.

An important aspect of working with algebraic expressions is recognizing the structure of these expressions. For instance, a binomial is an expression with two terms, and squaring it uses a specific pattern or formula as mentioned previously. Familiarity with different forms of algebraic expressions helps in identifying the most effective methods to simplify or manipulate them according to the given problem.

Simplifying Expressions

The process of simplifying expressions is aimed at making them easier to understand and work with. This involves combining like terms and using algebraic rules to streamline the expression. In the context of our binomial squaring example, once we apply the binomial square formula, we move to the simplification stage.

With \( x^2 + 2 * x * -3 + (-3)^2 \), we combine the terms where possible. The middle term, \( 2 * x * -3 \), simplifies to \( -6x \) and the term \( (-3)^2 \), simplifies to \( 9 \). Thus, the expression \( x^2 - 6x + 9 \) is the simplest form of the original binomial squared. It is important to follow the order of operations and careful arithmetic to ensure accuracy in the simplification process.