Question 1045791
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.