Question

Tell whether the statement is always, sometimes, or never true. Explain. The LCD of two rational expressions will have a degree greater than or equal to that of the denominator with the higher degree.

Short answer

The statement is always true. The LCD of two rational expressions will always have a degree that is greater than or equal to that of the denominator with the highest degree, because the degree of the LCD is the maximum degree amongst the denominators.

Step-by-step solution

Understand the statement

The statement suggests that the Lowest Common Denominator (LCD) of two rational expressions always has a degree that is greater than or equal to the degree of the denominator with the higher degree. A rational expression is a fraction in which the numerator and/or the denominator are polynomials.

Analyze LCD

In finding the LCD of two rational expressions, identify the denominators of the expressions and find their Least Common Multiple (LCM). In algebra, the degree of the LCD is the maximum degree among the denominators.

Conclusion

Based on the above analysis, the claim is valid. This is because the degree of the LCD of the two rational expressions is, at least, equal to the degree of the denominator with the highest degree. It is also worth noting that it is not necessarily greater since it depends on the denominators of the given rational expressions. Therefore, the given statement is always true.

Key concepts

Rational Expressions

Rational expressions are like regular fractions, but instead of numbers, they involve polynomials. You create a rational expression by placing a polynomial in both the numerator and the denominator. These expressions are called "rational" because they are ratios of two polynomials.
Understanding how to work with rational expressions is key in algebra, particularly when adding, subtracting, multiplying, or dividing them.
Key points to remember about rational expressions include:

• If the numerator or the denominator of a rational expression is zero, the expression is undefined.
• Just like with numeric fractions, you can simplify rational expressions by canceling common factors from the numerator and the denominator.
• Addition and subtraction require a common denominator, just as they do with regular fractions.

Learning to manipulate rational expressions effectively, particularly in seeking a common denominator, will allow you to solve complex algebraic problems more easily.

Degree of Polynomials

The degree of a polynomial is a basic concept that simply refers to the highest power of the variable in the polynomial. The degree tells us a lot about the behavior and characteristics of the polynomial.
In a polynomial like \(5x^3 + 2x^2 + x + 7\), the degree is 3, because \(x^3\) is the term with the highest exponent.
Observations about polynomial degrees include:

• The degree helps determine the number of solutions or roots the polynomial may have.
• A higher degree suggests a more complex curve when graphed.
• Knowing the degree is essential when performing operations like finding the LCD, as it helps predict the complexity of the resulting expression.

Understanding the degree of the polynomials within rational expressions is crucial in operations like finding the lowest common denominator.

Least Common Multiple (LCM)

The Least Common Multiple (LCM) is an essential arithmetic and algebraic tool employed to simplify expressions and solve equations. For rational expressions, the LCM of the denominators is used to find a common denominator, which is crucial when adding or subtracting these expressions.
Essentially, the LCM is the smallest multiple that is a multiple of two or more numbers (or polynomials in the context of rational expressions).
Key insights into dealing with LCM include:

• The LCM of polynomials involves taking the highest power of all terms that appear in each polynomial.
• When finding the LCM of numbers, it often involves listing the multiples of each number and identifying the smallest one they share.
• In algebra, knowing how to determine the LCM ensures you can manipulate expressions to have a common denominator, allowing for more straightforward addition and subtraction.

By mastering the concept of the LCM, you can more effectively work with rational expressions and understand the degree of complexity involved in their manipulation.