Problem 79 - Intermediate Algebra

Question

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{7 x-7 y}{x^{2}-y^{2}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{7}{x+y}\).

Step 1: Factor the Numerator

The numerator is \(7x - 7y\). The common factor in the terms \(7x\) and \(7y\) is \(7\). Factoring out \(7\), we get \(7(x - y)\).

Step 2: Recognize Denominator as a Difference of Squares

The denominator is \(x^2 - y^2\), a difference of squares. The formula for factoring a difference of squares is \(a^2 - b^2 = (a - b)(a + b)\).

Step 3: Factor the Denominator

Applying the difference of squares formula, we can factor the denominator \(x^2 - y^2\) as \((x - y)(x + y)\).

Step 4: Simplify the Expression

We now have: \[\frac{7(x - y)}{(x - y)(x + y)}\]The \((x - y)\) terms in the numerator and denominator can cancel each other out, simplifying the expression to \(\frac{7}{x + y}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components called factors. These factors, when multiplied together, give back the original polynomial. Recognizing common factors within polynomial expressions is essential. For instance, in the numerator \(7x - 7y\), both terms contain the common factor \(7\). Extracting this common factor simplifies the expression to \(7(x - y)\). This simplification is crucial for making the entire expression easier to work with and for subsequent simplification steps. Factoring polynomials can involve other techniques such as grouping, using the quadratic formula, or special forms like the difference of squares.

Difference of Squares

The difference of squares is a special algebraic pattern that applies to expressions in the form \(a^2 - b^2\). This can be factored into \((a - b)(a + b)\). Recognizing this pattern can greatly simplify algebraic expressions that fit this form. For the denominator \(x^2 - y^2\), it perfectly fits the difference of squares pattern. By applying this formula, you factor \(x^2 - y^2\) into \((x - y)(x + y)\). This technique is incredibly useful because it helps transform seemingly complex expressions into simpler ones that are easier to work with, facilitating the simplification process.

Canceling Common Factors

After factoring both the numerator and the denominator, the next step is to check if there are any common factors that can be canceled. Canceling common factors is like reducing a fraction to its simplest form. In the expression \(\frac{7(x - y)}{(x - y)(x + y)}\), the \((x - y)\) appears in both the numerator and denominator. These terms can be canceled out, simplifying the expression to \(\frac{7}{x + y}\). This cancellation simplifies calculations and helps achieve the expression's most basic form, which is easier to understand and apply in further computations.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions. Simplifying these fractions involves factoring and canceling common terms to reduce the fraction to its simplest form. In our given expression, \(\frac{7(x - y)}{(x - y)(x + y)}\), you simplify by factoring and then canceling the common terms \((x-y)\), bringing the fraction to \(\frac{7}{x + y}\). Working with algebraic fractions follows similar principles as numerical fractions but requires careful factoring and simplification of algebraic terms to maintain balance and correctness in the equation.