Just as a quadratic equation may have two real roots, so a cubic equation has possibly three. But unlike a quadratic equation which may have no real solution, a cubic equation always has at least one real root.
Similar remarks hold for cubic equations when the coefficients are all real. If the discriminant is positive, there are three distinct real roots. If the discriminant is negative, there is one real root and a complex conjugate pair of complex roots.
Nature of the roots
As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.
A cubic polynomial is of the form p(x) = ax3 + bx2 + cx + d. The values of 'x' that satisfy the equation p(x) = 0 are called roots of cubic polynomial p(x). Since the degree of p(x), the cubic equation p(x) = 0 can have a maximum of 3 roots.
Therefore we need −a3+3ab2+c<0 if the cubic is to have three positive roots. (In the diagram above the y-intercept is positive and you can see that the cubic has a negative root.) We determined earlier the condition for the cubic to have three distinct real roots, namely c2<4b6.
This is due to the fact that imaginary roots come in pairs. If a+bi is a root, then so is a-bi. Since a cubic equation has exactly 3 roots, then at least one must be a real number. A cubic equation can have three complex roots if the coefficients are complex.
The nature of roots of all cubic equations is either one real root and two imaginary roots or three real roots.
In a cubic polynomial number of zeros = 3 Why because degree of the polynomial = 3.
The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic function is monotonic.
The degree of the function indicates how many zeros the given function will have, therefore, the cubic function, since the degree is three, can have at most three zeros and it is not possible to have more than three.
In the first case, the quartic can have either 0, or 2, or 4 real roots; in the second case it can have either 0 or 2 real roots.
A cubic function is a polynomial function of degree 3. So the graph of a cube function may have a maximum of 3 roots. i.e., it may intersect the x-axis at a maximum of 3 points. Since complex roots always occur in pairs, a cubic function always has either 1 or 3 real zeros.
Therefore, a quadratic equation cannot have more than two roots.
So actually a cubic polynomial with real coefficients can only have 1 or 3 real roots, but not 2.
Every real number has a unique real cube root, and every nonzero complex number has three distinct cube roots.
Cubic Functions:
A cubic function is a power function with a degree power of 3. The domain of a cubic function is all real numbers because the cubic function is a polynomial function, which are continuous curves. The graph of f(x) = x3 is shown below.
A cubic function can also have two local extreme values (1 max and 1 min), as in the case of f(x) = x3 + x2 + x + 1, which has a local maximum at x = −1 and a local minimum at x = 1/3.
Because cubic polynomials have an odd degree, their end behaviors go in opposite directions, and therefore all cubic polynomials must have at least one real root.
A cubic polynomial will have 3 zeroes since its highest power (or degree) is 3.
That's no coincidence. The Fundamental Theorem of Algebra states that the degree of a polynomial is the maximum number of roots the polynomial has. A third-degree equation has, at most, three roots.
On the page Fundamental Theorem of Algebra we explain that a polynomial will have exactly as many roots as its degree (the degree is the highest exponent of the polynomial).
A quadratic equation of the form ax2+bx+c can have zero, one, or two distinct real roots; This is because the roots can be either real, complex or one root with multiplicity two.
One other thing worth mentioning: by the Intermediate Value Theorem, all cubics with real coefficients must have at least 1 real root (giving another reason such a cubic can't have 3 non-real complex roots).