Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Error Analysis A student claims that \((x-3)(x+3)=(x-3)^{2}\) Describe and correct the student's error.

Step-by-Step Solution

Answer

The student misunderstood the formula for the difference of squares. The correct equation when applying the difference of squares formula should be \((x-3)(x+3)=x^2 - 9\), not \((x-3)^{2}\).

Step 1: Identify the Error

The student is trying to use the formula \((a-b)^{2}= a^{2}-2ab+b^{2}\), while, in fact, should have used the formula \(a^2 - b^2 = (a - b)(a + b)\) for the difference of squares.

Step 2: Correct the Mistake

Let's correctly apply the differences of two squares formula. With \(a = x\) and \(b = 3\), we have \((x)^2 - (3)^2\), which simplifies to \(x^2 - 9\). This confirms that \((x-3)(x+3)=x^2 - 9\), not \((x-3)^{2}\).

Step 3: Check the Correction

We can check our correction by expanding \((x - 3)(x + 3)\). Doing so gives \(x^2 + 3x - 3x - 9\), which simplifies to \(x^2 - 9\), thus confirming our correction.

Key Concepts

Polynomials

Polynomials are expressions that consist of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents on the variables. The power of the polynomial is determined by the highest exponent in the expression. For example, in the expression \( x^2 - 9 \), we have a polynomial of degree 2 because the highest exponent is 2.

• The term \( x^2 \) is a polynomial term where the variable \( x \) is raised to the second power.
• The expression \(-9\) is a constant term, which is a polynomial of degree 0.

When working with polynomials, it's important to understand how different operations affect the terms within them. Knowing how to expand and simplify polynomials is key to solving algebraic problems efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They are foundational elements in algebra, as they allow us to represent numbers in a general form. In the problem at hand, \((x - 3)(x + 3)\) is an example of an algebraic expression. This particular expression involves multiplication and is structured to follow specific algebraic rules.

• Algebraic expressions can be simplified using algebraic identities, such as the difference of squares.
• They can also be rearranged using basic operations to achieve desired forms.

Understanding and manipulating algebraic expressions is crucial for solving equations and making sense of mathematical problems. It requires recognizing patterns and applying formulae correctly, as demonstrated in the provided exercise.

Difference of Squares

The difference of squares is a specific algebraic identity used to simplify expressions where two squared terms are subtracted. It follows the pattern \( a^2 - b^2 = (a - b)(a + b) \). This pattern is evident in the exercise where the expression \((x - 3)(x + 3)\) is incorrectly equated to \((x - 3)^2\).

• The correct application of the difference of squares helps in reducing errors that arise from misapplication of polynomial identities.
• Recognizing this identity simplifies expressions and prevents mistakes, such as assuming a square where there is none.

In algebra, ensuring the right identity is used is crucial, as it can significantly affect the results and interpretations of mathematical problems. Correctly applying the difference of squares to the given expression yields \( x^2 - 9 \), demonstrating its importance in simplifying and understanding algebraic expressions.