Problem 1 - Elementary and Intermediate Algebra

Question

Fill in the blanks. A quotient of two polynomials, such as \(\frac{x^{2}+x}{x^{2}-3 x},\) is called a ____ expression.

Step-by-Step Solution

Answer

The expression is called a rational expression.

Step 1: Identify the Structure

Examine the given expression \( \frac{x^{2}+x}{x^{2}-3x} \). It is made up of two polynomial expressions, one in the numerator (\( x^2 + x \)) and one in the denominator (\( x^2 - 3x \)).

Step 2: Define Rational Expression

In mathematics, when a quotient involves polynomials where one polynomial is divided by another, it is specifically termed as a 'rational expression.' This is because such expressions follow the form of ratios or fractions where both the numerator and denominator are polynomials.

Step 3: Fill in the Blank

Since the expression \( \frac{x^{2}+x}{x^{2}-3x} \) fits the definition identified in Step 2, you fill the blank with the term 'rational.' Therefore, the full sentence now reads: 'A quotient of two polynomials, such as \( \frac{x^{2}+x}{x^{2}-3x} \), is called a rational expression.'

Key Concepts

Understanding Polynomials

Polynomials are fundamental in algebra and involve expressions consisting of variables and coefficients. They appear in many forms and can vary in complexity. A polynomial may look like a simple expression such as \( x^2 + 3x + 2 \), where:

• Each term is a product of a coefficient and a variable raised to a non-negative integer power.
• The degree of the polynomial is determined by the highest power of the variable in the expression.

Polynomials can be added, subtracted, multiplied, and divided. This gives rise to expressions known as rational expressions when one polynomial is divided by another. The operations with polynomials follow specific rules that help maintain the structure and integrity of the expressions.

Identifying the Numerator

In a rational expression like \( \frac{x^2 + x}{x^2 - 3x} \), the numerator is the part above the division line. It represents the dividend in a division process.

• For our example, the numerator is \( x^2 + x \).
• This polynomial is quadratic, highlighting that the degree of the polynomial is 2, as 2 is the highest power of \( x \).

This placement is crucial because it forms one part of the ratio, and any operations or simplifications we perform start here. Understanding the numerator helps us see where the expression increases or decreases based on values of \( x \).

Decoding the Denominator

The denominator in any fraction or rational expression, such as \( \frac{x^2 + x}{x^2 - 3x} \), is the expression written below the division line. It acts as the divisor.

• In our example, the denominator is \( x^2 - 3x \).
• Like the numerator, it is also a polynomial, affecting the domains of the function it helps to create.

It is important to note that the denominator cannot be zero because division by zero is undefined. This affects the possible values that \( x \) can take, known as the domain of the rational expression. Always examine the denominator closely for restrictions on the values of variables.

Understanding the Quotient

The term 'quotient' refers to the result of a division. When working with rational expressions, such as \( \frac{x^2 + x}{x^2 - 3x} \), the expression itself represents a quotient of two polynomials.

• This means that the rational expression looks at how the numerator polynomial divides by the denominator polynomial.
• In algebra, simplifying or manipulating these expressions often involves factoring to reduce the expression into simpler terms.

The concept of a quotient is integral in understanding rational expressions because it highlights the relationship between the components of the expression. It's important to comprehend not just how to calculate a quotient but also what it symbolizes in terms of both numbers and algebraic function behaviors.