Can a polynomial have rational coefficients?

Any polynomial with rational coefficients can be made into a polynomial with integer coefficients by multiplying through by the product of the denominators of the coefficients (or the LCM of those denominators).

When can you not use rational root theorem?

any rational root fully reduced would have to have a numerator that divides evenly into 1 and a denominator that divides evenly into 2. Hence the only possible rational roots are ±1/2 and ±1; since neither of these equates the polynomial to zero, it has no rational roots.

What are the important things to consider when finding the rational roots of a given polynomial?

rational root theorem, also called rational root test, in algebra, theorem that for a polynomial equation in one variable with integer coefficients to have a solution (root) that is a rational number, the leading coefficient (the coefficient of the highest power) must be divisible by the denominator of the fraction and …

What does the rational root theorem and Descartes rule of signs indicate about the zeros of a polynomial function?

Descartes’ rule of sign is used to determine the number of real zeros of a polynomial function. It tells us that the number of positive real zeros in a polynomial function f(x) is the same or less than by an even numbers as the number of changes in the sign of the coefficients.

Can a polynomial with rational coefficients have irrational roots?

The irrational root theorem may be stated as follows: The irrational root theorem states that if the irrational sum of a plus the square root of b is the root of a polynomial with rational coefficients, then a minus the square root of b, which is also an irrational number, is also a root of that polynomial.

Can a polynomial have more roots than its degree?

The only polynomial with real or complex coefficients with more roots than its degree is the zero polynomial, . Every number is a root of the zero polynomial. Over the complex numbers, the Fundamental Theorem of Algebra states that a nonzero polynomial of degree has exactly complex roots if you count multiplicities.

How do you find the integer solution of a polynomial?

Integer solutions of a polynomial function theorem says: If a polynomial function. + a 1 x + a 0 = 0 with integer coefficients has an integer solution, a ≠ 0 , then that solution is the divisor of free coefficient . As an addition to this theorem, for every whole number k, number is a divisor of . Example 1.

How does rational root theorem and factor theorem helps in solving polynomial equation?

The rational roots theorem is a very useful theorem. It tells you that given a polynomial function with integer or whole number coefficients, a list of possible solutions can be found by listing the factors of the constant, or last term, over the factors of the coefficient of the leading term.