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How many unique roots will a third degree polynomial function have?
It can have 1, 2 or 3 unique roots.
At most how many unique roots will a third-degree polynomial have?
A third-degree equation has, at most, three roots. A fourth-degree polynomial has, at most, four roots. APEX 2021
At most how many unique roots will a fifth-degree polynomial have?
5, Using complex numbers you will always get 5 roots.
Is it true that the degree of polynomial function determine the number of real roots?
Sort of... but not entirely. Assuming the polynomial's coefficients are real, the polynomial either has as many real roots as its degree, or an even number less. Thus, a polynomial of degree 4 can have 4, 2, or 0 real roots; while a polynomial of degree 5 has either 5, 3, or 1 real roots. So, polynomial of odd degree (with real coefficients) will always have at least one real root. For a polynomial of even degree, this is not guaranteed. (In case you are interested about the reason for the rule stated above: this is related to the fact that any complex roots in such a polynomial occur in conjugate pairs; for example: if 5 + 2i is a root, then 5 - 2i is also a root.)
How many real roots do we have if the polynomial equation is in degree six?
Such an equation has a total of six roots; the number of real roots must needs be even. Thus, depending on the specific equation, the number of real roots may be zero, two, four, or six.
