Problem 6 - Technical Mathematics with Calculus

Question

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

Step-by-Step Solution

Answer

The factored form of the expression \(9x^2 - y^2\) is \((3x + y)(3x - y)\).

Step 1: Identify the Type of Polynomial

Recognize that the given expression, \(9x^2 - y^2\), is a difference of squares where \(9x^2\) is the square of \(3x\) and \(y^2\) is the square of \(y\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula, \(a^2 - b^2 = (a + b)(a - b)\), to factor the expression. In this case, \(a = 3x\) and \(b = y\).

Step 3: Write the Factored Form

Substitute \(3x\) for \(a\) and \(y\) for \(b\) into the formula to obtain the factored form of the expression as \((3x + y)(3x - y)\).

Key Concepts

Difference of Squares

Understanding the 'difference of squares' is vital for mastering algebraic factorization. It refers to an expression where one perfect square is subtracted from another. The general form of such an expression is \(a^2 - b^2\), where \(a\) and \(b\) represent any algebraic terms. This identity is special because it can be factored into the product of two binomials: \(a^2 - b^2 = (a + b)(a - b)\).

Recognizing a difference of squares in polynomial expressions allows for straightforward factorization. For instance, in the exercise \(9x^2 - y^2\), \(9x^2\) is a perfect square since \(3x\) multiplied by itself gives \(9x^2\), and similarly for \(y^2\), \(y\) times \(y\) is \(y^2\). By identifying the difference of squares, one can then apply the corresponding formula to simplify the expression.

Polynomial Factoring Step by Step

Factoring polynomials can initially seem daunting, but breaking it down into steps helps demystify the process. Here’s a step-by-step guide:

• Identify the type of polynomial and determine if there are any common factors to all terms.
• For a difference of squares, recognize the form \(a^2 - b^2\).
• Use the appropriate factoring formula, such as \(a^2 - b^2 = (a + b)(a - b)\) for a difference of squares.
• Substitute the identified terms (square roots of the perfect squares) back into the formula.
• Write the expression as a product of binomials or other polynomials, if possible.

Applying these steps to the given exercise, we first identify it as a difference of squares expression, apply the corresponding formula, and substitute \(a = 3x\) and \(b = y\) to reach the factored form: \(9x^2 - y^2 = (3x + y)(3x - y)\).

This systematic approach ensures that nothing is overlooked and makes polynomial factoring more approachable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In the context of the exercise, \(9x^2 - y^2\) is an algebraic expression consisting of two terms. Each term is made up of constants and variables raised to powers—the constants being \(9\) and \(1\) for \(9x^2\) and \(y^2\), respectively, and the variables being \(x\) with an exponent of \(2\) and \(y\), also with an exponent of \(2\).

The structure of algebraic expressions dictates the methods we use for simplification, including factoring. Algebraic expressions can also include sums, differences, products, quotients, and powers of variables and constants. Understanding how these components interact, and how operations like factoring apply to them, is foundational to solving algebraic problems.