Is it possible for a polynomial function of degree 3 with integer coefficient to have no real zeros explain your answer?

No. Roots that are not real must come in conjugate pairs, a+bi and a-bi, where a and b are real numbers, and so since a polynomial of degree 3 has 3 roots, only a maximum of one pair (aka 2 roots) can be not real, meaning there has to be at least 1 real root.

What is the polynomial of the smallest degree with integer coefficients and with zeros 3?

Answer. Required polynomial = x³ – 5x² + 8x – 6.

What is the polynomial of the smallest degree with integer coefficients and with zero s3 1 I 1?

If 1+i is a root and the coefficients are real then the conjugate i.e. 1-i is also a root. Hence the equation will have 4 roots . Therefore the minimum degree of the polynomial will be 4.

What polynomial has a degree of 3?

Cubic function
Polynomial Functions

Degree of the polynomial	Name of the function
2	Quadratic function
3	Cubic function
4	Quartic function
5	Quintic Function

What is the polynomial equation of degree 3?

A cubic equation is an algebraic equation of third-degree. The general form of a cubic function is: f (x) = ax3 + bx2 + cx1 + d. And the cubic equation has the form of ax3 + bx2 + cx + d = 0, where a, b and c are the coefficients and d is the constant.

What is a polynomial with a degree of 3?

cubic polynomial
A cubic polynomial is a polynomial of degree equal to 3.

Can a 3rd degree polynomial function have no real zeros?

There does NOT exist a 3rd degree polynomial with integer coefficients that has no real zeroes. The fact that if a pure complex number (one that contains “i”) is a zero then guarantees its conjugate is also a zero implies that the third zero has to be without the imaginary unit i.

What is the smallest degree the polynomial could have?

5
The minimum possible degree is 5.

What is a zero degree polynomial?

A polynomial function of degree zero has only a constant term — no x term. If the constant is zero, that is, if the polynomial f (x) = 0, it is called the zero polynomial. If the constant is not zero, then f (x) = a0, and the polynomial function is called a constant function.

What is the degree of a polynomial?

POLYNOMIALS (Polynomials with Real Coefficients) Definition 1: A real polynomial is an expression of the form P(x) = anxn + an−1xn−1 + ···+a1x+a0. where n is a nonnegative integer and a0,a1,…,an−1,an are real numbers with an 6= 0. The nonnegative integer n is called the degree of P. The numbers a0,a1,…,an−1,an.

How do you find real polynomials with 0?

POLYNOMIALS (Polynomials with Real Coefficients) Definition 1: A real polynomial is an expression of the form P(x) = anxn + an−1xn−1 + ···+a1x+a0. where n is a nonnegative integer and a0,a1,…,an−1,an are real numbers with an 6= 0.

Can a polynomial have two rational roots and one irrational root?

However, it is not possible for the polynomial p to have two rational roots r 1, r 2 and one irrational one z. That would imply that p = ( x − r 1) ( x − r 2) ( x − z), and clearly, the expanding this polynomial shows that the coefficient at x 2 is − z − r 1 − r 2. This number is rational only if z is also rational.

Is it possible to construct rational coefficients from Vieta’s formula?

To the first one of your queries the answer is – No, it’s not possible to construct if you want all coefficients to be rational. As, from Vieta’s formula we have sum of roots of a polynomial f ( x) = a x 3 + b x 2 + c x + d equals to − b / a, which is rational as you want all a, b, c, d to be integers.