Explain why 3x^{2} can be said to be in both standard form and factored form.
The Correct Answer and Explanation is :
The expression (3x^2) can indeed be viewed as both in standard form and factored form, depending on the context in which it is analyzed.
Standard Form: In algebra, a polynomial is typically expressed in standard form when it is written as a sum of terms in descending powers of the variable. For a quadratic expression, the standard form is represented as (ax^2 + bx + c), where (a), (b), and (c) are constants, and (a) is non-zero. In the case of (3x^2), it fits this definition as it can be considered a quadratic polynomial where (a = 3), (b = 0), and (c = 0). Thus, it is expressed purely in terms of (x^2) and lacks linear and constant terms, still adhering to the structure of a polynomial.
Factored Form: On the other hand, factored form refers to representing a polynomial as a product of its factors. For (3x^2), we can factor it as (3 \cdot x^2). Here, the number (3) acts as a constant factor, while (x^2) indicates the variable part. This representation highlights that the expression consists of the coefficient (3) multiplied by (x) squared, showcasing its roots and behavior more explicitly. The term (x^2) can be further factored into (x \cdot x), demonstrating that the expression consists of the repeated root (x = 0) with a multiplicity of two.
In conclusion, (3x^2) serves as an example of a polynomial that can simultaneously be in standard form and factored form. The standard form emphasizes its classification as a quadratic polynomial, while the factored form reveals its multiplicative structure and roots. This dual representation is vital in algebra, as it facilitates different approaches to solving equations, graphing, and understanding the behavior of polynomial functions.