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Chapter 12: Problem 2

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

Short Answer

Expert verified
True

Step by step solution

01

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.
02

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.
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} 2 x+y =-4 \\ -2 y+4 z =0 \\ 3 x -2 z=-11 \end{array}\right. $$

Theater Revenues A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for \(\$ 150,\) main seats for \(\$ 135,\) and balcony seats for \(\$ 110 .\) If all the seats are sold, the gross revenue to the theater is \(\$ 64,250\). If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is \(\$ 56,750 .\) How many of each kind of seat are there?

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} x-y+z=-4 \\ 2 x-3 y+4 z=-15 \\ 5 x+y-2 z=12 \end{array}\right. $$

Investments Kelly has \(\$ 20,000\) to invest. As her financial planner, you recommend that she diversify into three investments: Treasury bills that yield \(5 \%\) simple interest, Treasury bonds that yield \(7 \%\) simple interest, and corporate bonds that yield \(10 \%\) simple interest. Kelly wishes to earn \(\$ 1390\) per year in income. Also, Kelly wants her investment in Treasury bills to be \(\$ 3000\) more than her investment in corporate bonds. How much money should Kelly place in each investment?

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. What is the amount that results if \(\$ 2700\) is invested at \(3.6 \%\) compounded monthly for 3 years?
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