Problem 10 - Algebra Form and Function

Question

Which of the expressions are polynomials in \(x ?\) If an expression is not a polynomial in \(x,\) what rules it out? $$ \sqrt{2 x}-x^{3}+x^{5} $$

Step-by-Step Solution

Answer

Answer: No, the expression \(\sqrt{2x} - x^3 + x^5\) is not a polynomial in x because one of its terms, \(\sqrt{2x}\), has an exponent of \(\frac{1}{2}\) on x, which is not a positive integer nor zero.

Step 1: Check if the expression has a finite number of terms

The given expression has three terms: \(\sqrt{2x}\), \(-x^3\), and \(x^5\). Since there are finitely many terms, the expression passes the first rule.

Step 2: Check if the exponents on x are positive integers or zero

In the terms \(-x^3\) and \(x^5\), the exponents on x are positive integers: 3 and 5, respectively. However, in the term \(\sqrt{2x}\), x has an exponent of \(\frac{1}{2}\), which is not a positive integer nor zero. Hence, the given expression fails the second rule.

Step 3: Check if the coefficients of x are real numbers

For all three terms, the coefficients of x are real numbers: \(\sqrt{2}\) for \(\sqrt{2x}\), \(-1\) for \(-x^3\), and \(1\) for \(x^5\). The expression passes the third rule. Since the given expression fails the second rule (having positive integer exponents), it is not a polynomial in x.

Key Concepts

Polynomial Rules

Polynomials are mathematical expressions that follow a specific set of rules. These rules help us determine whether a given expression qualifies as a polynomial. Understanding these rules is essential for identifying polynomials correctly.

• A polynomial must have a finite number of terms. Each term can include constants and variables, but there cannot be infinitely many terms.
• The exponents of the variable in each term must be non-negative integers (0, 1, 2, ...). This means no term can have a variable raised to a negative or fractional power.
• Coefficients, the constant factors in front of the variables, must be real numbers. These can include any real number, whether they're integers, fractions, or irrational numbers like \(\sqrt{2}\).

Let's consider an example: the expression \(\sqrt{2x} - x^3 + x^5\). We note that it has three terms, adhering to the rule of a finite number of terms. However, it doesn't qualify as a polynomial because one term, \(\sqrt{2x}\), has an exponent of \(\frac{1}{2}\), which is a fractional number, thereby violating the second rule.

Exponents in Algebra

Exponents are a way of indicating that a number should be multiplied by itself a certain number of times. In algebra, dealing with exponents in expressions is crucial for constructing and understanding polynomials.For an expression to be a polynomial, the exponents attached to any variable must be non-negative integers. Here’s why:

• Intuitive Calculations: Non-negative integer exponents lead to straightforward arithmetic and algebraic operations. These operations include addition, subtraction, multiplication, and polynomial division.
• Smooth Functions: When exponents are non-negative integers, the resulting polynomial forms a smooth and continuous function which is easy to analyze and graph.
• Positive Powers: Polynomials don’t allow variables to be raised to negative or fractional powers, as they complicate calculations and lead to functions that aren't functions in some domains.

In the case of \(\sqrt{2x} - x^3 + x^5\), the term \(\sqrt{2x}\) includes an exponent of \(\frac{1}{2}\), which is a non-integer, and thus, hinders the expression from being classified as a polynomial.

Real Coefficients

In any polynomial, the coefficients, which are the numerical factors that multiply the variable terms, must be real numbers. Real numbers include all the numbers without imaginary components, spanning from irrational numbers like \(\sqrt{2}\) to rational numbers, integers, and zero.Here’s why real coefficients are a necessity:

• Broad Applicability: Real numbers cover a wide range of numeric values used in most everyday and scientific calculations. This makes polynomials with real coefficients more broadly applicable.
• Ease of Computation: Using real coefficients ensures that calculations involving the polynomial remain within real numbers, fostering easier arithmetic, algebraic manipulations, and applicable for a wide range of analysis and application.

In the provided expression \(\sqrt{2x} - x^3 + x^5\), we notice that all coefficients are real: \(\sqrt{2}\) for \(\sqrt{2x}\), \(-1\) for \(-x^3\), and \(1\) for \(x^5\). Having real coefficients fulfills one of the essential rules for classifying polynomials, but doesn't compensate for the other rules it's failing.