Problem 120 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the expression is a polynomial. If it is, write the polynomial in standard form. $$2 x^{2}-2 x^{4}-x^{3}+\sqrt{2}$$

Step-by-Step Solution

Answer

The given expression is a polynomial and the polynomial in the standard form is \(-2 x^{4} - x^{3} + 2 x^{2} + \sqrt{2}\)

Step 1: Identify whether the expression is a polynomial or not

A polynomial is an algebraic expression which consists of variables and coefficients. The given expression is \(2 x^{2}-2 x^{4}-x^{3}+\sqrt{2}\). It doesn't involve any operations that disqualify it from being a polynomial (such as division by a variable, square roots of a variable, or any trigonometric functions involving the variable). So, the given expression is a polynomial.

Step 2: Write the polynomial in standard form

The standard form of a polynomial is to write it in the order of decreasing powers. Here, we rearrange the terms of the polynomial \(2 x^{2}-2 x^{4}-x^{3}+\sqrt{2}\) to match this format: \(-2 x^{4} - x^{3} + 2 x^{2} + \sqrt{2}\).

Key Concepts

Standard Form of Polynomials

When dealing with polynomials, it's crucial to understand what the standard form is. This form is a way to write polynomials so that they are neat and consistent. A polynomial in standard form is arranged in descending order of the exponents of the variable.
For example, if we have a polynomial like \(2x^2 - 2x^4 - x^3 + \sqrt{2}\), we should rearrange it to have powers of \(x\) go from the highest to the lowest. This would give us \(-2x^4 - x^3 + 2x^2 + \sqrt{2}\).
This format not only looks nice, but it also makes doing calculations with the polynomial easier.

Characteristics of Polynomials

Polynomials are essential to algebra, and they come with some specific characteristics. Understanding these will help identify if an expression is truly a polynomial.
Here are key traits of polynomials:

• They consist only of variables raised to whole number exponents and their coefficients, which can be any real number.
• The operations involved are only addition, subtraction, and multiplication. No divisions by variables are allowed.
• They cannot have negative exponents, fractions, or square roots involving variables, nor can they include any other complex operations like trigonometric functions.

Identifying these traits in an expression helps clarify whether it is a polynomial. In our example, \(2x^2 - 2x^4 - x^3 + \sqrt{2}\), meets all these criteria, which confirms it is a polynomial.

Arranging Polynomial Terms

Once you determine that you have a polynomial, the next step is understanding how to arrange its terms properly. Arranging means ordering the terms by the degree of each term.
Here are steps to arrange a polynomial:

• First, identify the degree of each term, which is the highest power of \(x\) in that term. For example, in \(-2x^4\), the degree is 4.
• Write down the term with the highest degree first, followed by the others in decreasing order of their exponents.
• Keep any constant terms, which have an implicit degree of 0, at the end.

Applying this to our example, \(2x^2 - 2x^4 - x^3 + \sqrt{2}\), we rearrange it as \(-2x^4 - x^3 + 2x^2 + \sqrt{2}\), because \(-2x^4\) has the highest degree.