Problem 9 - 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? $$ \sqrt{2} x-x^{2}+x^{4} $$

Step-by-Step Solution

Answer

If not, provide a reason. Answer: Yes, the expression \(\sqrt{2} x-x^{2}+x^{4}\) is a polynomial in \(x\), because all the powers of \(x\) in the expression are non-negative integers.

Step 1: Expression given

We are given the expression: \(\sqrt{2} x-x^{2}+x^{4}\).

Step 2: Identify the terms

We can see that the given expression has three terms: 1. \(\sqrt{2} x\) 2. \(-x^{2}\) 3. \(x^{4}\)

Step 3: Check the powers of \(x\)

In order to be a polynomial, each term must have a non-negative integer power of \(x\). Let's check the powers of \(x\) in each term: 1. \(\sqrt{2} x\) has a power of \(x^1\) 2. \(-x^{2}\) has a power of \(x^2\) 3. \(x^{4}\) has a power of \(x^4\)

Step 4: Conclusion

All the powers of \(x\) in the given expression are non-negative integers. Therefore, the given expression \(\sqrt{2} x-x^{2}+x^{4}\) is a polynomial in \(x\).

Key Concepts

Non-Negative Integer Powers

When working with polynomials, it's essential to understand that the powers of the variable must always be non-negative whole numbers. This means that the exponents you see on the variable (often denoted as \(x\) in expressions) should be 0, 1, 2, and so on. For example, in the expression \(x^2 + 3x + 4\), the powers of \(x\) are 2, 1, and 0, which are all non-negative integers.

This rule helps to keep polynomials straightforward and ensures they can be graphed easily. If you come across a negative or fractional power, like \(x^{-1}\) or \(x^{1/2}\), the expression is not considered a polynomial. Such cases don't fit into the standard form of polynomial equations.

• A polynomial example: \(2x^3 + x - 5\)
• Not a polynomial: \(x^{0.5} + 3\)

Terms of a Polynomial

The building blocks of a polynomial are its terms. A polynomial is comprised of one or more terms, which are separated by '+' or '-' signs. Each term is a product of a coefficient (a number) and the variable raised to a power. For instance, in the polynomial \(3x^3 + 2x^2 - x + 7\), the terms are \(3x^3\), \(2x^2\), \(-x\), and \(7\).

These terms can have varying degrees, which refers to the power of the variable in each term. In any polynomial, identifying and understanding each term is crucial for performing operations like addition, subtraction, or finding roots.

• Example of terms in a polynomial: In \(4x^2 - 3x + 5\), the terms are \(4x^2\), \(-3x\), and \(5\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication). They serve as the foundation for various mathematical concepts, including polynomials. While all polynomials are algebraic expressions, not all algebraic expressions qualify as polynomials.

For example, an expression like \(3x^2 + 2x + 1\) is both a polynomial and an algebraic expression. However, \(\frac{1}{x} + x^2\) is only an algebraic expression because it includes a term, \(\frac{1}{x}\), with a negative power. Algebraic expressions may also include operations such as division, and not all forms align with the definition of a polynomial.

• Example of an algebraic expression that isn't a polynomial: \(x + \frac{1}{x}\)
• Example of a polynomial: \(5x^4 - 2x + 9\)