Problem 2

Let . You must answer all the parts of this question without fully calculating determinants by cofactor expansion.

(a). Is a Polynomial in , what is the Highest Degree Term in the Polynomial (after the Polynomial is Simplified by Collecting like terms)?

The highest degree is three.

(b). Let Be a Polynomial such that and . Can Be Factored?

For and as they switch signs there must be some value where thus there is some factorization which includes .

(c). Prove that Has Factors and

For to be factorable by there must some value of where

For since the determinant must be zero and thus the determinant of must be 0 as well.

For since , the determinant must be zero and thus the determinant of must be 0 as well.

Since for both and their determinant is 0 there must be some factorization for some constant as the top term is