Can a fourth degree polynomial have no real zeros?

Can a fourth degree polynomial have no real zeros?

So to construct a quartic with no Real zeros, start with two pairs of Complex conjugate numbers. Or you could simply start with any quartic polynomial with positive leading coefficient, then increase the constant term until it no longer intersects the x axis.

What degree polynomial has no real zeros?

degree 0
For non-zero complex polynomials, this turns out to be true in general and follows directly from the fundamental theorem of algebra. Indeed, a polynomial of degree 0 takes on the form c0 , where c0≠0 c 0 ≠ 0 , and thus has no zeros.

How many zeros does a degree 4 polynomial have?

This function is zero for only one value of x , namely x=0 . So in one sense you could say that it has one zero. By the Fundamental Theorem of Algebra, any quartic equation in one variable has exactly 4 roots – counting multiplicity. In this particular example, it has one root of multiplicity 4 , namely x=0 .

Can a 4th degree function have 3 real zeros?

A fourth degree polynomial has four roots. Non-real roots come in conjugate pairs, so if three roots are real, all four roots are real. If there are only three distinct real roots, one root is duplicated. Therefore, your polynomial factors as p(x)=(x−a)2(x−b)(x−c).

How do you prove a polynomial has no zeros?

If the discriminant of the equation < 0 then the given polynomial has no zeros.

What’s a 4 degree polynomial?

In algebra, a quartic function is a function of the form. where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form. where a ≠ 0.

Can a degree 4 polynomial with real coefficients have exactly 0 real roots?

A polynomial of even degree can have any number from 0 to n distinct real roots. A polynomial of odd degree can have any number from 1 to n distinct real roots.

Is it possible to have exactly 3 real zeros Why?

Any degree 3 polynomial with real coefficients has at least one real zero. In fact any polynomial of odd degree with real coefficients has at least one real zero. So, by the Intermediate Value Theorem, somewhere in between we have p(x)=0.