Question

True or False The rational expression \(\frac{5 x^{2}-1}{x^{3}+1}\) is proper.

Short answer

True

Step-by-step solution

Identify the Rational Expression

The given rational expression is \( \frac{5 x^{2}-1}{x^{3}+1} \). A rational expression involves a numerator and a denominator, both of which are polynomial expressions.

Determine Degrees of Polynomials

The degree of the numerator (\(5x^2 - 1\)) is the highest exponent of the variable \(x\), which is 2. The degree of the denominator (\(x^3 + 1\)) is the highest exponent of the variable \(x\), which is 3.

Compare the Degrees

To determine if the rational expression is proper, compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the rational expression is proper.

Conclusion

Since the degree of the numerator (2) is less than the degree of the denominator (3), the rational expression \( \frac{5 x^{2}-1}{x^{3}+1} \) is proper.

Key concepts

Polynomial Degrees

When working with rational expressions, it's important to understand polynomial degrees. The degree of a polynomial is the highest power of the variable in that polynomial. For example, in the polynomial expression \(5x^2 - 1\), the degree is 2 because the highest exponent of \(x\) is 2. Similarly, in \(x^3 + 1\), the degree is 3 because the highest exponent of \(x\) is 3.

Remember, identifying the degree is crucial when determining if a rational expression is proper. A rational expression is proper if the degree of the numerator is less than the degree of the denominator.

Proper Fractions

Proper fractions in algebra are similar to proper fractions in arithmetic. For a rational expression to be considered a proper fraction, the degree of the polynomial in the numerator must be less than the degree of the polynomial in the denominator.

Let's revisit the given example: \( \frac{5 x^{2}-1}{x^{3}+1} \). Here, the degree of the numerator (5x^2 - 1) is 2, whereas the degree of the denominator (x^3 + 1) is 3. Since 2 is less than 3, this rational expression is proper.

Understanding proper fractions helps in simplifying and performing operations with rational expressions. It also ensures that we know how to categorize and work with these expressions effectively.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (addition, subtraction, multiplication, division). Rational expressions, like the ones we are discussing here, are specific types of algebraic expressions where one polynomial is divided by another.

For instance, in \( \frac{5 x^{2}-1}{x^{3}+1} \), both \(5x^2 - 1\) and \(x^3 + 1\) are polynomials. The entire expression is an algebraic expression because it involves fundamental operations on polynomials.

To effectively work with algebraic expressions:

• Simplify whenever possible.
• Factorize both the numerator and the denominator where applicable.
• Understand the properties of polynomials involved in the expression.

Mastering these concepts will make solving problems involving rational expressions much easier and more intuitive.