Polynomial with a degree of 3,with real coeficient,and with a root of z=1-i. Help!

Question

Need help with the problem given above. Thanks

Answer 1

When the coefficients are real, the Conjugate Root Theorem tells us that if 1 - i is a root, so is its conjugate, i + 1. So you have have two roots. The number of roots must equal the degree of the polynomial, which is 3. Hence we need one more root. The final root must be a real number because if it were complex, its conjugate would also be a root and we'd have four roots instead of three. In the absence of additional information in the problem statement, it appears you are free to choose any real root. So let's pick 1. Use the factored form of a polynomial. With degree three we need three factors:

p(x) = a·(x-p)(x-q)(x-r)

where a is a constant and p, q, r are the roots. Choose a = 1.

p(x) = (x-(1-i))(x-(1+i))(x-1)

Multiply it out if you want it in standard form. There are an infinite number of polynomials that have roots of 1-i and 1+i since we could choose any real number for the last root and any real number for the constant a.