Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-52: Simplify the difference quotient \(\frac{f(x)-f(a)}{x-a}\) if \(x \neq a\). \(f(x)=x^{3}-2\)

Step-by-Step Solution

Answer

The simplified difference quotient is \( x^2 + xa + a^2 \).

Step 1: Identify the Function Components

First, let's determine what the problem is asking us to simplify. We are given the function \( f(x) = x^3 - 2 \). The difference quotient is defined as \( \frac{f(x) - f(a)}{x - a} \). We will need to find both \( f(x) \) and \( f(a) \).

Step 2: Calculate \( f(a) \)

Using the given function \( f(x) = x^3 - 2 \), substitute \( a \) into the function to find \( f(a) \). This gives us \( f(a) = a^3 - 2 \).

Step 3: Set Up the Difference Quotient

Substitute \( f(x) \) and \( f(a) \) into the difference quotient formula. This results in \( \frac{x^3 - 2 - (a^3 - 2)}{x-a} \). Simplify the numerator by distributing the negative sign: \( x^3 - 2 - a^3 + 2 \).

Step 4: Simplify the Numerator

The expression \( x^3 - 2 - a^3 + 2 \) simplifies to \( x^3 - a^3 \) because the \( -2 \) and \( +2 \) cancel each other out.

Step 5: Factor the Simplified Numerator

Recognize that \( x^3 - a^3 \) is a difference of cubes. It can be factored as \((x-a)(x^2 + xa + a^2)\).

Step 6: Cancel Common Factors

Substitute the factored form into the difference quotient: \( \frac{(x-a)(x^2 + xa + a^2)}{x-a} \). Since \( x eq a \), cancel the \( x-a \) terms from the numerator and the denominator. You are left with \( x^2 + xa + a^2 \).

Key Concepts

Polynomial Functions

A polynomial function is an expression composed of variables and coefficients, consisting of one or more terms connected by addition or subtraction. One key characteristic of polynomial functions is that the variables appear only to whole number powers. For example, in the polynomial function \( f(x) = x^3 - 2 \), we identify a single term, \( x^3 \), which is a simple power of \( x \).

This class of functions is significant in mathematics due to its unfaltering properties across calculations, such as the ability to determine derivatives, integrals, and other critical points easily. Understanding polynomial functions lays the groundwork for more complex algebraic concepts and provides a versatile tool for mathematical applications.

• Terms are built from powers of variables, like \( x^3 \).
• Polynomials can be single-term (monomials), two-term (binomials), or even three-term (trinomials).
• A polynomial's degree is the highest power of the variable present.

Difference of Cubes

The expression \( x^3 - a^3 \) represents the difference of cubes. This is a special technique in algebra used to factor expressions of that form. The formula for factoring a difference of cubes is:
\[ x^3 - a^3 = (x-a)(x^2 + xa + a^2) \]

Understanding this identity can simplify a complex expression immediately, which is especially useful in calculus where simplification is often required for differentiation. It's vital to memorize the standard pattern for these expressions:

• Identify each perfect cube: \( x^3 \) and \( a^3 \).
• Apply the formula to factor it into a simple binomial times a trinomial.
  

Knowing how to recognize and factor the difference of cubes can be a powerful tool in solving algebraic equations.

Factoring

Factoring is a process of breaking down an expression into products of simpler expressions or numbers. In algebra, this often means breaking down polynomials into products of functions that are irreducible over the integers. This approach simplifies complicated algebra problems significantly and allows for easier manipulation of expressions.

For example, when you factor \( x^3 - a^3 \) as \( (x-a)(x^2 + xa + a^2) \), it becomes more manageable. Factoring is key in solving equations and simplified computation.

• Identify terms that can be factored out.
• Apply known factoring formulas (like the difference of cubes).
• Reduce the complexity of your expressions.

Mastering factoring can greatly enhance your ability to simplify expressions and solve algebraic equations efficiently.

Simplification in Algebra

Simplification in algebra is essential for making complex expressions and equations easier to work with. The goal is to condense expressions into their simplest form while retaining their original value. This often involves combining like terms, factoring expressions, and canceling common factors in fractions.

For the difference quotient discussed in the original exercise, simplifying involved both recognizing a difference of cubes and factoring it accordingly. By simplifying, we went from a complex-looking fraction to a tidy polynomial expression, \( x^2 + xa + a^2 \), which is easier to work with.

• Combining like terms focuses on addition/subtraction within the expression.
• Factoring reduces terms using their common factors.
• Canceling terms in equations involves striking out equal terms from numerator and denominator.

Through simplification, algebra becomes less cumbersome and opens the pathway to solving for unknowns or optimizing expressions.