A cubic polynomial p(x) has leading coefficient 1, all real coefficients, and p(3 - 4i) = 0. If p(0) = −52, find p(x).
Call the one real root , r The other two roots are 3 + 4i and 3 -4i
(x - r) ( x - (3 +4i)) ( x - (3 -4i) = 0
(x - r) (x^2 - 6x + 25) = 0
x^3 - 6x^2 + 25x
- rx^2 + 6rx - 25 r
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x^3 - (6+r)x^2 + (25 + 6r) - 25 r = 0
Since p(0) = -52
Then -25r= -52 ⇒ r = 52/25 = the real root
So
p(x) = x^3 - (6+52/25)x^2 + (25 +6*52/25)x - 52 simplify
p(x) = x^3 - (202/25)x^2 + (937/25)x - 52