Problem 2 - Technical Mathematics with Calculus

Question

Factor completely.$$x^{2}-9$$

Step-by-Step Solution

Answer

\((x - 3)(x + 3)\)

Step 1: Identify the Type of Polynomial

Recognize the given polynomial, \(x^{2}-9\), as a difference of squares, since it can be written as \(x^{2} - 3^2\).

Step 2: Write as a Difference of Squares

Express the polynomial as a difference of two perfect squares: \((x)^2 - (3)^2\).

Step 3: Apply the Difference of Squares Formula

Use the formula \(a^2 - b^2 = (a - b)(a + b)\) to factor the polynomial. Let \(a = x\) and \(b = 3\).

Step 4: Factor the Polynomial

The polynomial \(x^{2} - 9\) becomes \((x - 3)(x + 3)\) after factoring.

Key Concepts

Difference of Squares

Understanding the concept of the difference of squares is essential in factoring polynomials. This special case occurs when a polynomial can be expressed as the subtraction of two perfect squares. In mathematical terms, it follows the pattern of a^2 - b^2, where both a and b are real numbers or algebraic expressions. When confronted with such a pattern, you can always factor it using the formula (a - b)(a + b). This works because if you were to expand the factored form, you would return back to the original expression a^2 - b^2.

Polynomial Factoring Steps

Factoring polynomials is like breaking down a mathematical phrase into its essential 'words'. To do this effectively, you need to follow certain steps.

• Identify the type of polynomial: Check if the polynomial is a difference of squares, a perfect square trinomial, or another form that can be factored.
• Write in its distinct form: For the difference of squares, express it clearly as a^2 - b^2.
• Apply the suitable factoring formula: Depending on the polynomial type, use the appropriate formula to factor it.
• Factor completely: Ensure all factors are irreducible and cannot be factored further. This might include factoring out a greatest common factor (GCF) first.

By walking through these steps, you can simplify complex algebraic expressions into products of simpler ones, making them more manageable to work with.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers, where letters represent unknown values. They range from simple forms, like x + 5 or a^2, to more complex polynomials, like 3x^2 + 2x - 1. The beauty of algebra lies in the flexibility of these expressions to represent a variety of real-world situations. When you're working with algebraic expressions, especially in the case of factoring, it's crucial to recognize patterns and common factors. This process not only simplifies calculations but also enhances your understanding of mathematical structures and their applications.