Problem 80 - Algebra A Combined Function

Question

Explain how to write a rational expression in lowest terms.

Step-by-Step Solution

Answer

Identify and cancel common factors in the numerator and denominator to simplify the expression.

Step 1: Understand the Rational Expression

A rational expression is a fraction where both the numerator and the denominator are polynomials. The expression in question must be in fraction form to proceed.

Step 2: Factor the Numerator and Denominator

Identify and factor all the polynomials in the numerator and the denominator of the rational expression. For example, if the expression is \( \frac{x^2 - 4}{x^2 - x - 6} \), factor it to \( \frac{(x-2)(x+2)}{(x-3)(x+2)} \).

Step 3: Determine Common Factors

Observe the factored form of both the numerator and denominator to identify any common factors. In the example, (x+2) is a common factor.

Step 4: Cancel Common Factors

Cancel out the common factors from both the numerator and the denominator. This will simplify the expression. For our example, cancel (x+2) to get \( \frac{x-2}{x-3} \).

Step 5: Verify the Lowest Terms

Ensure that there are no more common factors between the numerator and the denominator. The remaining expression is now in lowest terms. \( \frac{x-2}{x-3} \) does not have any more common factors.

Key Concepts

Polynomial

A polynomial is a fundamental concept in mathematics. It's an expression that involves variables raised to various powers and is composed of terms that are either added or subtracted. Each term in a polynomial consists of a coefficient multiplied by a variable raised to a non-negative integer exponent. For example, in the expression \(x^2 - 4\), \(x^2\) and \(-4\) are terms that make up the polynomial.
Polynomials can have multiple terms:

• A monomial has one term (e.g., \(7x^3\)).
• A binomial has two terms (e.g., \(x^2 - 4\)).
• A trinomial consists of three terms (e.g., \(x^2 - x - 6\)).

Polynomials are used in various mathematical contexts, including the expressions seen in rational expressions.

Factoring

Factoring is a crucial process in simplifying polynomials, where you express a polynomial as a product of simpler polynomials. It is akin to breaking down a number into its prime factors. For instance, factoring the polynomial \(x^2 - 4\) involves expressing it as a product \((x-2)(x+2)\).
Here are the steps to factor a polynomial:

• Look for the greatest common factor (GCF) in all terms.
• Apply special factoring formulas, like the difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).
• Factor trinomials using techniques like trial and error or grouping.

Factoring aids in both simplifying expressions and solving polynomial equations by revealing potential solutions.

Numerator and Denominator

In a rational expression, the numerator and denominator are both polynomials, where the numerator is the expression above the division line and the denominator is below it.
Understanding their roles is crucial as:

• The numerator, when divided by the denominator, gives the value of the rational expression at points where the denominator is not zero.
• The denominator should never be zero, as division by zero is undefined.

In our example \(\frac{x^2 - 4}{x^2 - x - 6}\), \(x^2 - 4\) is the numerator and \(x^2 - x - 6\) is the denominator. Factoring allows us to further investigate how these interact and simplify if necessary.

Lowest Terms

Writing a rational expression in lowest terms is the process of simplifying the expression so that the numerator and the denominator have no common factors other than 1. Simplifying to lowest terms makes the expression easier to work with, and clearer to analyze.
To achieve this:

• Factor both the numerator and the denominator completely.
• Identify and cancel out any common factors.
• Verify no common factors remain in the simplified expression.

In doing so, not only does the expression become more concise, but it also aids in accurately comparing the relations of different rational expressions.