- Identify the degree of each term: \(7, 5, 3, 1,\) and \(0\).
- Arrange the terms from the highest to the lowest degree: \(3y^7 + 2y^5 - 4y^3 + y + 12\).
- Compare the result with the given options to find the match.
- The correct option is \(\boxed{c}\).
Explanation
- Analyzing the Degrees of the Terms
To write the polynomial \(2 y^5+y+3 y^7-4 y^3+12\) in standard form, we need to arrange its terms in descending order of their exponents. The terms and their corresponding degrees are:
- \(3y^7\) has a degree of 7.
- \(2y^5\) has a degree of 5.
- \(-4y^3\) has a degree of 3.
- \(y^1\) (written as \(y\)) has a degree of 1.
- \(12\) (written as \(12y^0\)) has a degree of 0.
- Ordering the Terms in Descending Order
Now, we arrange these terms according to their degrees in decreasing order: 7, 5, 3, 1, 0.
- Starting with the highest degree, we have \(3y^7\).
- The next highest degree is 5, giving us \(2y^5\).
- Following that is degree 3, which is \(-4y^3\).
- Then degree 1, which is \(y\).
- Finally, the constant term of degree 0 is \(12\).
Combining these terms in this sequence gives the standard form of the polynomial: \(3 y^7 + 2 y^5 - 4 y^3 + y + 12\).
- Comparing with Options and Conclusion
Comparing our result \(3 y^7 + 2 y^5 - 4 y^3 + y + 12\) with the provided options:
A\(12+y-4 y^3+2 y^5+3 y^7\)
B\(12-4 y^3+3 y^7+2 y^5+y\)
C\(3 y^7+2 y^5-4 y^3+y+12\)
D\(y+2 y^5+3 y^7-4 y^3+12\)
Our result matches option (c). Thus, the correct order of the polynomial is \(3 y^7 + 2 y^5 - 4 y^3 + y + 12\). The final answer is \(\boxed{c}\).
Examples
Polynomial ordering is a fundamental skill in algebra used for simplifying complex expressions and solving equations. For example, when an engineer models the trajectory of a rocket using a high-degree polynomial, they must arrange it in standard form to efficiently calculate roots (the moments when the rocket reaches certain altitudes or impacts the ground). Standardizing the polynomial helps in applying systematic methods like synthetic division or identifying the leading behavior of the function, which is critical for accurate modeling and design.