Question 1210413: Find a degree 3 polynomial with real coefficients having zeros 1 and 4i and a lead coefficient of 1. Write P in expanded form. Be sure to write the full equation, including P(x) =.
Answer by ikleyn(52957)   (Show Source):
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Find a degree 3 polynomial with real coefficients having zeros 1 and 4i and a lead coefficient of 1.
Write P in expanded form. Be sure to write the full equation, including P(x) =.
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If a polynomial with real coefficients has a root which is a complex number,
then this polynomial has another root which is the conjugate complex number.
It is a general property of polynomials with real coefficients.
In our case, it means that the seeking polynomial has the roots 4i, -4i and 1.
Hence, the seeking polynomial is
P(x) = (x-1)*(x-4i)*(x-(-4i)) = (x-1)*(x^2+16) = x^3 - x^2 + 16x - 16.
ANSWER. P(x) = x^3 - x^2 + 16x - 16.
Solved.
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