Given the roots of a polynomial equation with rational coefficients, state all other known roots of the equation

Question

Given the roots of a polynomial equation with rational coefficients, state all other known roots of the equation

4, 3 - i

Answer 1

We don't know the degree of the polynomial, so any number of roots is possible.

Given the roots, 4, 3+i and 3-i, we have the factors x - 4, and x² - 6 + 10, giving us a polynomial of:

x³ - 10x² + 34x - 40

This could be multiplied by any number of real or complex factor pairs to get a higher-order polynomial

Answer 2

Hello Alyssa,

When we know that a zero of a polynomial is a complex number, a+ bi. We are assured that there is another complex zero, and that it is the conjugate of the known zero, a - bi.

For your problem, the zeros are as follow:

4, 3 - i, and 3 + i

Regards,

Anita A.