SOLUTION: A polynomial of degree 4 with real coefficients has -2i and -5+7i as two of its zeros. What are the two remaining zeros for this polynomial? I have no clue where to start. All

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Question 901852: A polynomial of degree 4 with real coefficients has -2i and -5+7i as two of its zeros. What are the two remaining zeros for this polynomial?

I have no clue where to start. All the problems I have done so far give me the polynomial.Please help
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
complex roots come in pairs.

the solution of -2i is one of the roots.
it's conjugate of +2i is the other root.

the solution of -5+7i is one of the roots.
it's conjugate of -5-71 is the other root.

that's a total of 4 roots which is what you should have since the degree of the equaton is fourth degree.

remember:

complex roots always come in pairs.

a complex root is a root with a real part and an imaginary part.

the real part of -2i is equal to 0

that root is really 0 - 2i and 0 + 2i

any time you see just an imaginary part, you can be sure that the real part is equal to 0 and the complex root is really 0 +/- the imaginary part.

the real part of 0 is not normally shown.

if we multiply the roots of this equation, you will be able to see the quadratic equation itself.

you will not, however, see the graph crossing the x-axis since the graph will only cross the x-axis where the real roots are.

your equation is found by multiplying the factors together.

the roots of the equation are shown as:

-2i
+2i
-5-7i
-5+7i

the factors of the equation are:


y = (x - 2i) * (x + 2i) * (x - 5 - 7i) * (x - 5 + 7i)

i multiplied those factors out and got the following equation:

y = x^4 - 10x^3 + 78x^2 - 40x + 296

basically, you multiply the complex roots in pairs.

the i part will cancel out.

if i did this right, you should get the following:

(x-2i) * (x+2i) results in x^2 + 4

(x-5-7i) * (x-5+7i) results in x^2 - 10x + 74

you then multiply (x^2 + 4) * (x^2 - 10x + 74) to get x^2 - 10x^3 + 78x^2 - 40x + 296.

the graph of the equation looks like this:

since all the roots of the equation are complex, the equation does not cross the x-axis.

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