Problem 39 - Introductory Algebra for College Students

Question

Find and simplify. \(\frac{f(x)-f(h)}{x-h}\) $$f(x)=x^{2}-1$$

Step-by-Step Solution

Answer

The simplified expression is \(x+h\).

Step 1: Substitute for \(f(x)\) and \(f(h)\)

First, replace \(f(x)\) with \(x^{2}-1\) and \(f(h)\) with \(h^{2}-1\) in the expression. This results in \(\frac{x^{2}-1 - (h^{2}-1)}{x-h}\).

Step 2: Simplify the numerator

Next, simplify the numerator by cancelling out the -1 with the +1. This results in \(\frac{x^{2} - h^{2}}{x - h}\).

Step 3: Factor the numerator

Since \(x^{2} - h^{2}\) is a difference of squares, factor it as \((x - h)(x + h)\). Now the expression looks like \(\frac{(x - h)(x + h)}{x - h}\).

Step 4: Simplify the expression

Cancel the \(x-h\) terms from the numerator and the denominator. This simplifies the expression to \(x+h\).

Key Concepts

Algebraic Simplification

In algebra, simplification involves reducing an expression to its most basic form while retaining its original value. Simplification makes mathematical expressions easier to understand and work with, especially in complex calculations.

When simplifying algebraic expressions, look for like terms to combine, constants that can be canceled out, and the use of distributive, associative, or commutative properties to rearrange the terms. In our exercise, the simplification process began with substituting the given functions into the rational expression and then cancelling the numerical constants. The final simplification involved recognizing and factoring a difference of squares and then reducing common factors.

Factoring Polynomials

Factoring polynomials is a critical skill in algebra. It involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give the original polynomial. Recognizing patterns, such as the difference of squares in our problem, is a key to successful factoring.

For example, the difference of squares is a special polynomial pattern where two terms are squared and subtracted, and it can be factored into \(a^2 - b^2 = (a - b)(a + b)\). This pattern was utilized in the exercise to factor \(x^2 - h^2\) into \(x - h)(x + h)\), allowing us to then simplify the expression by canceling out common factors in the numerator and denominator.

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator, the denominator, or both. Simplifying rational expressions involves factoring polynomials and canceling out common factors from the numerator and denominator, as shown in the exercise.

It's important to understand that a rational expression cannot be simplified by canceling terms that are not factors. For instance, in the expression \(\frac{x^2 - h^2}{x - h}\), you can cancel \(x - h\) after factoring, as it is a factor of the numerator. Simplifying rational expressions makes them easier to evaluate and can also prepare them for further algebraic operations such as addition, subtraction, multiplication, or division of rational expressions.