SOLUTION: A polynomial f(x) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f(x) as a product of linear and/or quadratic polynomials with real coeffi

Question 1113491: A polynomial f(x) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f(x) as a product of linear and/or quadratic polynomials with real coefficients that are irreducible over the set of real numbers.
7 + 5i, −1 + i; degree 4
Answer by ikleyn(52814) (Show Source): You can put this solution on YOUR website!
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At given condition (polynomial with real coefficients) each complex (non-real) root goes in pair with its conjugate.


So,   if  7+5i  is the root,  then  7-5i  is the root, too.

Also, if  -1+i  is the root,  then  -1-i  is the root, too.


Thus we have 4 roots  7+5i, 7-5i, -1+i, -1-i.


Hence, our polynomial is the product


p(x) = (x-(7+5i))*(x-(7-5i))*(x-(-1+i))*(x-(-1-i)) = 

     = ((x-7)-5i)*((x-7)+5i)*((x+1)-i)*((x+1)+i) = 

     = ((x-7)^2 - (5i)^2)*((x+1)^2 - i^2) =

     = (x^2 - 14x + 49 + 25)*(x^2 + 2x + 1 + 1) = 

     = (x^2 - 14x + 74)*(x^2 + 2x + 2).


It is the required expression.

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