SOLUTION: I am having the most difficult time figuring out how to format this into a polynomial. I know the highest exponent degree must be at 4, and that it also must include x^2 but that i

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Question 1045791: I am having the most difficult time figuring out how to format this into a polynomial. I know the highest exponent degree must be at 4, and that it also must include x^2 but that is all I can figure out. Any help would be much appreciated.

Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 4; Zeros -5+5i; 1 multiplicity 2
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Degree four means you will have something like %28x-a%29%28x-b%29%28x-c%29%28x-d%29 and the zeros are a, b, c, and d.

You are given two of the zeros, and one of them is repeated (multiplicity of 2).
%28x-%28-5%2B5i%29%29%28x-1%29%28x-1%29%28x-c%29 for some not yet finished value c.

To determine your last binomial factor, you must understand that Complex zeros occur in conjugate pair. This means, c=-5-5i.

Now your polynomial IN FACTORED FORM is %28x-%28-5%2B5i%29%29%28x-%28-5-5i%29%29%28x-1%29%5E2.

Now simplify and arrange the polynomial into general form.

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the equation is degree 4.
therefore you should have 4 roots.
the highest exponent will be 4.

let's look at what you have:

you are asked to Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 4; Zeros -5+5i; 1 multiplicity 2

your roots are, or should have been given as:

x = 5 plus or minus 5i,
x = 1 with a multiplicity of 2.

complex roots always come in pairs.

if a + bi is a root, then a - bi is also a root.

to find the factors, your set x equal to the root and then set the equation equal to 0.

for example:

x = 1 is set to 0 by doing the following:
subtract 1 from both sides of the equation to get x - 1 = 0
x - 1 is a factor.

they told you the root has a multiplicity of 2, therefore 2 of your factors are (x-1) * (x-1) which can also be written as (x-1)^2.

your complex roots come in pairs.

they should be 5 + 5i and 5 - 51.

set x = 5 + 5i
subtract 5 + 5i from both sides of the equation to get:
x - 5 - 5i = 0
that's one of the complex factors.

set x = 5 - 5i
subtract 5 from both sides of the equation and add 5i to both sides of the equation to get:
x - 5 + 5i = 0
that's the other of the complex factors.

your factors are (x-1)^2 * (x - 5 - 5i) * (x - 5 + 5i)

your equation will be:

(x-1)^2 * (x - 5 - 5i) * (x - 5 + 5i) = 0

multiply these factors out to get your equation.

(x - 5 - 5i) * (x = 5 + 5i) results in x^2 - 10x + 50

(x-1)^2 results in x^2 - 2x + 1

your equation would be:

(x^2 - 10x + 50) * (x^2 - 2x + 1) = 0

the result of that would be:

x^4 - 12x^3 + 71x^2 - 110x + 50 = 0

the general equation is y = x^4 - 12x^3 + 71x^2 -110x + 50.

to factor that equation, you set y = 0 to get what we had before as:

x^4 - 12x^3 + 71x^2 - 110x + 50 = 0

now it's in standard form for factoring.