When the polynomial is written in standard form, what are the values of the leading coefficient and the constant?
[tex]5 x+2-3 x^2[/tex]
A. The leading coefficient is 5, and the constant is 2.
B. The leading coefficient is 2, and the constant is 5.
C. The leading coefficient is -3, and the constant is 2.
D. The leading coefficient is 2, and the constant is -3.
Answer (1)
Rewrite the polynomial in standard form: − 3 x 2 + 5 x + 2 .
Identify the leading coefficient: − 3 .
Identify the constant term: 2 .
The leading coefficient is − 3 and the constant is 2 .
Explanation
Rewrite in Standard Form First, we need to rewrite the polynomial in standard form. Standard form means arranging the terms in descending order of their exponents. The given polynomial is 5 x + 2 − 3 x 2 . In standard form, this becomes − 3 x 2 + 5 x + 2 .
Identify Leading Coefficient Next, we identify the leading coefficient. The leading coefficient is the coefficient of the term with the highest degree. In the standard form − 3 x 2 + 5 x + 2 , the term with the highest degree is − 3 x 2 . Therefore, the leading coefficient is − 3 .
Identify Constant Term Finally, we identify the constant term. The constant term is the term without any variable. In the standard form − 3 x 2 + 5 x + 2 , the constant term is 2 .
Final Answer Therefore, the leading coefficient is − 3 and the constant is 2 .
Examples
Understanding polynomials and their coefficients is crucial in many areas, such as physics and engineering. For example, when analyzing the trajectory of a projectile, the equation describing its height as a function of time is a polynomial. The leading coefficient and constant term can tell us important information about the projectile's initial conditions and the forces acting upon it. By understanding these concepts, we can predict the behavior of the projectile and design systems that optimize its performance.
