Q: The polynomial given roots

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The "roots" of a polynomial are the solutions of the equation polynomial = 0. That is, any value which you can replace for "x", to make the polynomial equal to zero.

You forgot to copy the polynomial. However, the Fundamental Theorem of Algebra states that every polynomial has at least one root, if complex roots are allowed. If a polynomial has only real coefficients, and it it of odd degree, it will also have at least one real solution.

Rational roots

4

The real roots of what, exactly? If you mean a square trinomial, then: If the discriminant is positive, the polynomial has two real roots. If the discriminant is zero, the polynomial has one (double) real root. If the discriminant is negative, the polynomial has two complex roots (and of course no real roots). The discriminant is the term under the square root in the quadratic equation, in other words, b2 - 4ac.

Related questions

The given polynomial does not have factors with rational coefficients.

The "roots" of a polynomial are the solutions of the equation polynomial = 0. That is, any value which you can replace for "x", to make the polynomial equal to zero.

No integer roots. Quadratic formula gives 1.55 and -0.81 to the nearest hundredth.

A third degree polynomial could have one or three real roots.

You forgot to copy the polynomial. However, the Fundamental Theorem of Algebra states that every polynomial has at least one root, if complex roots are allowed. If a polynomial has only real coefficients, and it it of odd degree, it will also have at least one real solution.

Since no polynomial was given, no answer will be given.

Actually, the roots of a Hurwitz polynomial are in the left half of the complex plain, not on the imaginary axis. As for the reason, that is because the polynomial is DEFINED to be one that has that kind of roots.

In answering this question it is important that the roots are counted along with their multiplicity. Thus a double root is counted as two roots, and so on. The degree of a polynomial is exactly the same as the number of roots that it has in the complex field. If the polynomial has real coefficients, then a polynomial with an odd degree has an odd number of roots up to the degree, while a polynomial of even degree has an even number of roots up to the degree. The difference between the degree and the number of roots is the number of complex roots which come as complex conjugate pairs.

Rational roots

4, the same as the degree of the polynomial.

A third-degree equation has, at most, three roots. A fourth-degree polynomial has, at most, four roots. APEX 2021

There are none because the discriminant of the given quadratic expression is less than zero.