Two roots of a third degree polynomial function f(x) are -4 and 4. Which statement describes the number and nature of all
roots for this function?
* f(x) has three imaginary roots.
* f(x) has three real roots.
* f(x) has two real roots and two imaginary roots.
* f(x) has three real roots and one imaginary root.

Answer:

f(x) has three real roots.

Step-by-step explanation:

f(x) has three real roots.

A third degree polynomial function with real coefficients has exactly 3 roots.

Furthermore, all complex roots come in pairs, either there is none, or there are exactly two conjugates, which are in the form e1+i(e2) and e1-i(e2) where e1 and e2 are polynomial expressions.

This also means that a third degree polynomial with real coefficients has at least one real root, or three.

Since we are given two real roots for the given polynomial, the third root cannot be complex alone, so the only possibility is that there are three real roots.

If b is real, then all three roots are real.

If b is not real, then two of the roots are real (-4 and 4) and one is imaginary.

Polynomial is an equation written with terms of the form kx^n.

where k and n are positive integers.

There are quadratic polynomials and cubic polynomials.

Example:

2x³ + 4x² + 4x + 9 is a cubic polynomial.

4x² + 7x + 8 is a quadratic polynomial.

We have,

The polynomial function f(x) has degree 3 and two roots -4 and 4, we know that (x + 4) and (x - 4) are factors of the function.

To see why, note that when x = -4, the factor (x + 4) becomes zero, making the function zero.

Similarly, when x = 4, the factor (x - 4) becomes zero, making the function zero.

Now,

Since the function has degree 3 and two factors of degree 1, we can conclude that the third factor is also of degree 1.

This means that the function can be written in the form:

f(x) = a (x + 4) (x - 4) (x - b)

where a and b are constants.

We don't know the value of b, but we know that it is a real number (since the other two roots are real).

Thus, the three roots of the function are -4, 4, and b.

Now,

To determine whether the roots are all real or some are imaginary, we need to know the value of b.

If b is real, then all three roots are real.

If b is not real, then two of the roots are real (-4 and 4) and one is imaginary (b).

Thus,

If b is real, then all three roots are real.

If b is not real, then two of the roots are real (-4 and 4) and one is imaginary.

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