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Question

Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros

 

4+ i, 1

 

f(x)=

Answer 1

Since the polynomial has rational coefficients, complex roots occur in conjugate pairs.  So, since 4 + i is a root, 4 - i must also be a root.  

 

So a polynomial with smallest possible degree would be:                      

 

f(x) = (x - 1)[x - (4+i)][x - (4 - i)]

 

         = (x - 1)[(x-4) - i][(x-4) + i]

 

         = (x - 1)[(x-4)² - i²]

 

         = (x-1)(x² - 8x + 17)

 

         = x³ - 8x² + 17x - x² + 8x - 17

 

         = x³ - 9x² + 25x - 17

Answer 2

When we have a solution that is a complex number, the polynomial will have a degree of 2.  This polynomial is a parabola.  The reason for this is because we use the quadratic formula to find the zeros.  This formula has a square-root part and if the number under the square-root is negative, the roots will be complex numbers.  So the function is

 

f(x) = (x - 1)(x - 4 - i)

 

 

Expand f(x).

 

f(x) = x(x - 4 - 1) - 1(x - 4 - i)

 

f(x) = x² - 4x - x - x + 4 + i

 

f(x) = x² - 6x + i