The standard form of the given polynomial is f(x) = 2x^6 + 6x^5 + 12x^3 - 6x^2 - 6x + 4.
The given polynomial is f(x) = 2x^6 + 6x^5 + 12x^3 - 6x^2 - 6x + 4.
In standard form, a polynomial is written in descending order of powers of (x). Let's rearrange the terms to put the polynomial in standard form:
f(x) = 2x^6 + 6x^5 + 12x^3 - 6x^2 - 6x + 4
Reordering the terms in descending order of powers of (x), we get:
f(x) = 2x^6 + 6x^5 + 12x^3 - 6x^2 - 6x + 4
So, the standard form of the given polynomial is f(x) = 2x^6 + 6x^5 + 12x^3 - 6x^2 - 6x + 4.
Understanding the properties of the polynomial involves analyzing its factors and roots, which can be found by setting \(f(x)\) equal to zero. Techniques such as factoring, synthetic division, or numerical methods like the rational root theorem may be employed for this purpose.
The graph of the polynomial is expected to exhibit characteristics based on its degree and coefficients, showcasing trends such as oscillations, peaks, and troughs. The leading term (2x^6) implies that the graph will rise or fall rapidly as (|x|) increases.