Question

Is it possible to write two rational functions whose sum is a quadratic function? Justify your answer.

Short answer

Yes, it is possible to write a quadratic function as the sum of two rational functions when each of the rational functions are constants, linear or quadratics such that the total order upon addition does not exceed 2.

Step-by-step solution

Understand the nature of rational functions

Rational functions in general form appear as \( f(x) = p(x)/q(x) \). In fact, any polynomial function, including a quadratic function, can be viewed as a certain type of rational function where the denominator is simply 1.

Consider the sum of two rational functions

When adding two rational functions, the result could still be a rational function. If the denominator is not equal to 1, the sum would not be a quadratic function. However, if the denominator is equal to 1, then it would be a polynomial. The question is now if such a polynomial could be a quadratic one.

Explore Possibility of Resulting in a Quadratic Function

If the sum of these two rational functions is a quadratic function, this means that each of these rational functions do not have orders more than two. In other words, the rational functions are allowed to be constant (order 0), linear (order 1), or quadratic (order 2) themselves to be able to sum up to a quadratic.

Finalize the Conclusion

From the above discussion, it can be rightly said that it is possible to write two rational functions whose sum is a quadratic function. This is possible when each of these rational functions are either a constant, linear or quadratic, and when they are added, the highest order does not exceed 2.

Key concepts

Quadratic Function

A quadratic function is a type of polynomial function characterized by a specific form:

• The standard form is given by an expression: \( ax^2 + bx + c \).
• Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\).
• The graph of a quadratic function is a parabola, which can open upwards or downwards.
• The vertex of the parabola can be found using the formula \(x = -\frac{b}{2a}\).

Quadratic functions are fundamental in algebra and appear frequently in various mathematical problems. They can represent a wide range of real-world scenarios, from calculating areas to understanding projectile motion. Understanding their properties such as the vertex, axis of symmetry, and roots, can help solve these problems efficiently.

Polynomial Function

Polynomial functions are expressions that involve sums of powers of a variable. They have the general form:

• \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \).
• Here, \(a_n, a_{n-1}, \ldots, a_0\) are coefficients, and \(n\) is the degree of the polynomial, which is a non-negative integer.
• The highest power of the variable determines the degree of the polynomial.

Polynomials can be classified based on their degrees:

• Linear: Degree 1 (e.g., \(3x + 2\)).
• Quadratic: Degree 2 (e.g., \(x^2 - 4x + 4\)).
• Cubic: Degree 3 (e.g., \(2x^3 + x^2 - x + 1\)).

Polynomials are versatile and can model many types of functions, including rational, exponential, and even periodic functions in certain interpretations. The roots or solutions of polynomials are found by factoring or using methods like the Quadratic Formula for degree 2 polynomials.

Addition of Rational Functions

Rational functions are formed by dividing one polynomial by another. They take the form:

• \( f(x) = \frac{p(x)}{q(x)} \)
• Where \(p(x)\) and \(q(x)\) are polynomial functions.
• The function is defined for all values except where the denominator is zero.

When adding rational functions, the process involves finding a common denominator. Here's how it works:

• Identify the least common denominator (LCD) of the rational functions.
• Rewrite each function as an equivalent fraction with the LCD as the new denominator.
• Add the numerators of these equivalent fractions.
• Simplify, if possible, the resulting expression into a simpler form.

In some cases, the sum might result in a rational function having a numerator that simplifies to a polynomial. It is possible for this to be a quadratic polynomial if the highest degree of the combined result is 2. This typically happens when the individual fractions align to form a polynomial expression after addition.