Question

explain why the facts given are contradictory.

f(x) is a polynomial function of degree 4 whose coefficients are real numbers; three of its zeros are 2, 1 + 2i, and 1 - 2i. Explain why the remaining zero must be a real number.

Short answer

The fourth root should be a real one because if the fourth root is complex then its conjugate also becomes a root of the polynomial hence we have 5 roots and degree of polynomial becomes 5 which will be a contradiction

Step-by-step solution

Given information

We are given a statement

f(x) is a polynomial function of degree 4 whose coefficients are real numbers; three of its zeros are 2, 1 + 2i, and 1 - 2i. Explain why the remaining zero must be a real number.

Explanation

We are given that degree of polynomial is 4 and one root is 2 and other roots are 1+2i , 1-2i these are 3 roots the fourth root should be a real one because if the fourth root is complex then its conjugate also becomes a root of the polynomial hence we have 5 roots and degree of polynomial becomes 5 which will be a contradiction.