- Understand that standard form for a multivariate polynomial relies on lexicographical order, meaning terms are organized by descending powers of \(x\).
- Analyze the powers of \(x\) in each option: \(1, 3, 4, 7\); \(5, 2, 1, 0\); \(5, 1, 2, 0\); and \(0, 1, 9, 7\).
- Identify that only the sequence \(5, 2, 1, 0\) follows the descending order requirement.
- Conclude that the polynomial \(\boxed{18x^5 - 7x^2y - 2xy^2 + 17y^4}\) is the one in standard form.
Explanation
- Defining the Standard Form of a Polynomial
A polynomial is typically considered to be in standard form when its terms are arranged in a specific, consistent order. For a polynomial in multiple variables, such as \(x\) and \(y\), the most common convention is the lexicographical order. In this convention, we first order the terms by the exponent of the first variable (typically \(x\)) in descending order. If two terms have the same power of \(x\), we order them by the exponent of the next variable (\(y\)), also in descending order. We will evaluate the given options based on these criteria.
- Analyzing the Polynomials
Let's examine the powers of \(x\) in each of the given options:
- \(3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4\): The powers of \(x\) are \(1, 3, 4, 7\). This is an ascending sequence.
- \(18x^5 - 7x^2y - 2xy^2 + 17y^4\): The powers of \(x\) are \(5, 2, 1, 0\). This is a descending sequence.
- \(x^5y^5 - 3xy - 11x^2y^2 + 12\): The powers of \(x\) are \(5, 1, 2, 0\). This sequence is not monotonic.
- \(15 + 12xy^2 - 11x^9y^5 + 5x^7y^2\): The powers of \(x\) are \(0, 1, 9, 7\). This sequence is not monotonic.
- Identifying the Standard Form Polynomial
Based on the lexicographical ordering criterion, a polynomial is in standard form when the powers of the variable \(x\) are arranged in descending order. Our analysis shows that only the second polynomial, \(18x^5 - 7x^2y - 2xy^2 + 17y^4\), satisfies this requirement, as its terms have \(x\)-powers of \(5, 2, 1, 0\) respectively. Therefore, this polynomial is in standard form.
Examples
Standard form is essential in algebra for keeping expressions organized, much like how we organize files in folders on a computer or arrange items in a library. By keeping polynomials in a predictable order (like lexicographical order), it becomes much easier for mathematicians to perform operations such as addition, subtraction, or long division, as the like terms are automatically grouped or aligned. This practice is fundamental in simplifying complex algebraic expressions in physics, engineering, and computer science algorithms where symbolic manipulation is required.