SOLUTION: Find a polynomial f(x) of degree 3 with real coefficients and the following zeros: 4, 2 + i f(x)=___

SOLUTION: Find a polynomial f(x) of degree 3 with real coefficients and the following zeros: 4, 2 + i f(x)=____ thanks!

Question 1119625: Find a polynomial f(x) of degree 3 with real coefficients and the following zeros:
4, 2 + i
f(x)=____
thanks!
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

A polynomial function of degree

has exactly

zeros counting multiplicities. Further, complex zeros always appear in conjugate pairs. If

is a zero of a polynomial function, then

is also a zero of that polynomial function. And finally, if

is a zero of the function, then

is a factor of the function.

Therefore, the three factors of your 3rd degree polynomial function must be:

Multiply the three factors together and collect like terms to find your polynomial. Computational hint: Treat the two complex number factors as binomials where the complex numbers are constants (which they actually are) and recall that the product of two binomial conjugates is the difference of two squares. However, since you are squaring

which results in -1, you actually get the SUM of two squares. The result will be a quadratic polynomial which you can then multiply by the real number factor

.

John

My calculator said it, I believe it, that settles it

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