Problem 42 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$9 x^{2}-81 y^{2}$$

Step-by-Step Solution

Answer

The polynomial factors as \(9(x - 3y)(x + 3y)\).

Step 1: Identify Common Factors

The given polynomial is \(9x^2 - 81y^2\). We start by looking for any common factors in each term. Notice that both terms are divisible by 9.

Step 2: Factor Out the Common Monomial

Since 9 is common to both terms, factor 9 out of the polynomial: \(9(x^2 - 9y^2)\).

Step 3: Recognize the Difference of Squares

Observe the expression inside the parentheses: \(x^2 - 9y^2\). It is a difference of squares, as it fits the pattern \(a^2 - b^2\), where \(a = x\) and \(b = 3y\).

Step 4: Apply the Difference of Squares Formula

Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). Using this formula, factor \(x^2 - 9y^2\) as \((x - 3y)(x + 3y)\).

Step 5: Write the Complete Factorization

Combine the factored terms with the common factor extracted earlier: \(9(x - 3y)(x + 3y)\). This is the complete factorization of the original polynomial.

Key Concepts

Common Monomial Factor

When confronted with a polynomial that needs to be factored, always begin by searching for a common monomial factor. This involves identifying any number or variable that divides evenly into each term of the polynomial. In the case of the polynomial \(9x^2 - 81y^2\), observe the coefficients: both 9 in \(9x^2\) and 81 in \(-81y^2\) are divisible by 9.

Extracting this common factor simplifies the polynomial. As such, factoring out 9 transforms the expression into \(9(x^2 - 9y^2)\). This step simplifies the polynomial, making it easier to apply further factoring techniques. Always factor out the greatest common monomial first to streamline the process of complete factorization.

Difference of Squares

After factoring out the common monomial, examine the polynomial for patterns of special products, such as the difference of squares. A difference of squares is an expression that fits the structure \(a^2 - b^2\), where \(a\) and \(b\) are any expressions. These patterns can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).

For the expression inside the parentheses in \(9(x^2 - 9y^2)\), identify \(x^2 - 9y^2\) as a difference of squares. Here, \(a = x\) and \(b = 3y\), because \(9y^2\) can be rewritten as \((3y)^2\).

Apply the formula to factor \(x^2 - 9y^2\) into \((x - 3y)(x + 3y)\). Recognizing such patterns simplifies the factorization process significantly and is key in algebraic manipulation.

Complete Factorization

Complete factorization involves breaking the polynomial down into its most basic components or factors, such that multiplying these factors together yields the original polynomial. After extracting the common monomial factor and applying the difference of squares in the previous steps, the expression \(9(x^2 - 9y^2)\) has been transformed.

Utilizing the identified difference of squares, we factor \(x^2 - 9y^2\) as \((x - 3y)(x + 3y)\). Thus, the complete factorization becomes \(9(x - 3y)(x + 3y)\).

Complete factorization is crucial because it not only makes expressions easier to work with but is also essential for solving equations and simplifying algebraic expressions.