Problem 79 - Calculus Single Variable

Question

\- A real-valued function \(f\) of a real variable \(x\) is said to be algebraic if there is a polynomial \(p(u, v)\) with integer coefficients such that \(p(x, f(x))=0\) for all \(x .\) For example, \(f(x)=\) \(2 x+\sqrt{x^{2}+1}\) is algebraic because $$ \begin{aligned} p(x, f(x))=& 3 x^{2}+\left(2 x+\sqrt{x^{2}+1}\right)^{2} \\ &-4 x\left(2 x+\sqrt{x^{2}+1}\right)-1 \\ \equiv & 0 \end{aligned} $$ for \(p(u, v)=3 u^{2}+v^{2}-4 u v-1 .\) A function that is not algebraic is said to be transcendental. Find a polynomial that shows that the given expression is algebraic. $$ \sqrt[3]{x+\sqrt{x}} $$

Step-by-Step Solution

Answer

The polynomial \( p(x, y) = y^6 - 2xy^3 + x^2 - x \) shows the function is algebraic.

Step 1: Understand the Problem

We need to determine if the function \( f(x) = \sqrt[3]{x + \sqrt{x}} \) is algebraic. This means finding a polynomial \( p(u, v) \) such that \( p(x, f(x)) = 0 \).

Step 2: Express the Function

Let \( y = \sqrt[3]{x + \sqrt{x}} \). Then \( y^3 = x + \sqrt{x} \). We will work on eliminating the square root by squaring both sides.

Step 3: Eliminate the Square Root

First, express \( \sqrt{x} \) in terms of \( x \) and \( y \). We have \( y^3 - x = \sqrt{x}\). Square both sides to eliminate the square root: \( (y^3 - x)^2 = x \).

Step 4: Expand and Rearrange the Expression

Expanding gives \( y^6 - 2xy^3 + x^2 = x \). Rearrange to form a polynomial equation: \( y^6 - 2xy^3 + x^2 - x = 0 \).

Step 5: Identify the Polynomial

The required polynomial with integer coefficients is \( p(x, y) = y^6 - 2xy^3 + x^2 - x \), and this polynomial demonstrates that the function is algebraic.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables combined using only addition, subtraction, multiplication, and non-negative integer exponents. For example, a simple polynomial could be expressed as:

• Linear Polynomials: like \( ax + b \)
• Quadratic Polynomials: like \( ax^2 + bx + c \)
• Cubic Polynomials: like \( ax^3 + bx^2 + cx + d \)

These expressions can include multiple terms and are characterized by their degree, which indicates the highest power of the variable. Each term is composed of a coefficient (a number) and a combination of the variables. This allows these functions to represent a broad range of relationships within mathematical and real-world situations. Polynomials with integer coefficients are especially useful because they maintain the integrity of numbers without introducing fractions or decimals.

Transcendental Functions

Transcendental functions are the fancy cousins of polynomial functions! They cannot be expressed as a finite series of polynomial equations. Common examples include exponential, logarithmic, and trigonometric functions. A key characteristic of transcendental functions is that they go beyond the limitations of algebraic equations.

• Exponential functions like \( e^x \), where \( e \) is Euler's number, are not tied to the finite structure of polynomials.
• Trigonometric functions, such as \( \sin(x) \) and \( \cos(x) \), frequently appear in periodic contexts.
• Logarithmic functions, such as \( \log(x) \), are the inverse of exponential functions.

Transcendental functions play a significant role in higher mathematics and most scientific applications, where complex representations are needed to define growth, oscillations, and waves.

Integer Coefficients

In algebra, integer coefficients are a desirable property in polynomial equations. They are whole numbers, both positive and negative, including zero. For example, in the expression \( 3x^2 + 5x - 7 \), the coefficients \( 3, 5, \) and \(-7 \) are all integers.Using integer coefficients has several advantages:

• Simplicity in computation, since integers are easier to manage than fractions or decimals.
• Grants precision and clarity to an equation, omitting the complexities introduced by irrational numbers or fractions.
• Facilitates finding solutions to equations using integer operations.

In our exercise, the polynomial \( y^6 - 2xy^3 + x^2 - x \) is expressed entirely with integer coefficients. This makes it straightforward and ensures clear communication of mathematical relationships.

Elimination of Square Roots

When dealing with functions involving square roots, like \( \sqrt{x} \), the elimination of square roots is a useful technique. The goal is to remove the irrational component in order to form algebraic expressions.Follow these steps to eliminate square roots:

• Isolate the square root on one side of the equation if possible.
• Square both sides of the equation in order to eliminate the square root.
• Simplify the resulting equation, being careful to maintain equality.

For instance, in the given problem, we start with \( y^3 = x + \sqrt{x} \) and square both sides to obtain \( (y^3 - x)^2 = x \), successfully eliminating the square root. This transformation often results in a polynomial expression that can be more conveniently analyzed and solved, aiding in the determination of algebraic properties of the function.