Write a polynomial function with rational coefficients that has 6 possible rational zeros according to the rational zero theorem, but no actual rational zeros . You must show the possible rational zeros and the actual answers.

- Thread starter c_r_a_15
- Start date

Write a polynomial function with rational coefficients that has 6 possible rational zeros according to the rational zero theorem, but no actual rational zeros . You must show the possible rational zeros and the actual answers.

- Joined
- Nov 12, 2017

- Messages
- 3,023

Here's my thought: First decide on first and last coefficients that would give you 6 possible rational zeros. Then try filling in other coefficients.

To avoid too much trial and error, you might just make it a quadratic that has no real zeros at all!

I suspect that it might help to start at another place.

by the Fundamental Theorem of Algebra, a polynomial with real coefficents and degree n has exactly n complex roots (although some may be duplicated). Furthermore , if n is odd, at least one of those roots is real. So the simplest polynomial with six distinct real roots is of degree 6.

Moreover, if a polynomial of degree 2k has 2k real zeroes, then that polynomial can be factored into k quadratics, each of which has two real zeroes.

by the Fundamental Theorem of Algebra, a polynomial with real coefficents and degree n has exactly n complex roots (although some may be duplicated). Furthermore , if n is odd, at least one of those roots is real. So the simplest polynomial with six distinct real roots is of degree 6.

Moreover, if a polynomial of degree 2k has 2k real zeroes, then that polynomial can be factored into k quadratics, each of which has two real zeroes.

Last edited: