Problem 71 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Simplify the rational expression, if possible. \(\frac{x^2-4 x-5}{x^2+4 x-5}\)

Step-by-Step Solution

Answer

The simplified form of the rational expression is \(\frac{(x-5)(x+1)}{(x+5)(x-1)}\).

Step 1: Factor the Numerator and Denominator

Begin by factoring both the numerator and the denominator of the given rational expression. In this case, \(x^2-4x-5\) factors into \((x-5)(x+1)\) and \(x^2+4x-5\) factors into \((x+5)(x-1)\)

Step 2: Cancel out common factors

Next look for any common factors between the numerator and denominator which can be cancelled out. There are not any common factors, so the expression can't be further simplified.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill when working with algebraic expressions, particularly in simplifying rational expressions. It involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give you the original polynomial.

For example, in our exercise \(x^2-4x-5\), this quadratic polynomial can be factored by finding two numbers that multiply to -5 (the constant term) and add to -4 (the coefficient of the linear term). These numbers are -5 and +1, giving us the factors \(x-5\) and \(x+1\) respectively when employing factorization methods like foil or the area method.

Understanding how to factorize efficiently can prevent errors and lead to quicker simplification of rational expressions.

Rational Expression Simplification

Simplification of rational expressions means to reduce the expression to its simplest form. This can be done by identifying and cancelling any common factors in the numerator and denominator.

The process uses factoring skills and a strong grasp of algebraic division. For instance, if we had a common factor of \(x+1\) in both the numerator and the denominator, we would divide both by this term to simplify the expression.

However, in our problem, the numerator \(x^2-4x-5\) factors to \(x-5)(x+1)\), and the denominator \(x^2+4x-5\) factors to \(x+5)(x-1)\), showing no common factors to cancel out. Recognizing when a rational expression is already in its simplest form is just as important as being able to simplify it.

Canceling Common Factors

Canceling common factors pertains to the process where the same factor that appears in both the numerator and the denominator of a rational expression is divided out, effectively 'cancelling' it.

This can only be done when the numerator and denominator are fully factored and is akin to reducing fractions in basic arithmetic. For example, if we had the expression \(\frac{(x+2)(x-3)}{(x-3)(x+4)}\), the \(x-3\) term is a common factor and can be cancelled to simplify the expression to \(\frac{x+2}{x+4}\).

In the case of our exercise, as no common factors are present in the fully factored forms of the numerator and the denominator, no cancellation can take place. This highlights the importance of factorization as an initial step in simplification.