What is the equation telling us?

The equation is telling us that if we were to take 4 away from x, we would be left with 32x squared plus 9.

What is the highest possible degree of the polynomial?

The highest possible degree of the polynomial is 4.

What are the real roots of the polynomial?

There are four real roots of the polynomial equation. They are -3, -1, 1, and 3.

What are the imaginary roots of the polynomial?

The imaginary roots of the polynomial are -3i and 3i.

How do we solve for the roots of the polynomial?

There are a few different ways that we can solve for the roots of the polynomial. One way is to use the quadratic equation, which is:

-b +/- sqrt(b^2-4ac)/2a

In this equation, a is the coefficient of x^2, b is the coefficient of x, and c is the constant term. We can plug in the values from our polynomial to solve for the roots. Another way to solve for the roots is to use factoring. We can factor this polynomial by grouping:

(x^4+32x^2+9)=(x^2+9)(x^2+4)

From here, we can see that two of the roots are 3 and -1. The other two roots would be -3 and 1.

What is the nature of the roots of the polynomial?

The roots of the polynomial are real and equal to -3 and 1/3.

What is the meaning of the discriminant of the polynomial?

The discriminant of a polynomial is a quantity that can be used to determine the number and type of roots of the polynomial. If the discriminant is positive, the polynomial has two real roots and no complex roots. If the discriminant is zero, the polynomial has two real roots and two complex roots. If the discriminant is negative, the polynomial has four complex roots.

What are the conditions for the existence of real roots?

The equation above has four roots, two of which are imaginary. In order for an equation to have real roots, the discriminant (b^2-4ac) must be greater than or equal to zero. Therefore, the discriminant of this equation must be greater than or equal to zero in order for it to have real roots.

What are the conditions for the existence of imaginary roots?

Imaginary roots only exist when the discriminant, b2 – 4ac, is negative. In the equation above, the discriminant is equal to 32 – 4(9) = -104. Since the discriminant is negative, there are no real roots and the equation has imaginary roots.