Form a polynomial F(x) with real coefficients having the given degree and zeros

Degree: 4; zeros -5+3i; 4 multiplicity 2

Question

Form a polynomial F(x) with real coefficients having the given degree and zeros

Degree: 4; zeros -5+3i; 4 multiplicity 2

anonymous · Answer

ick

anonymous · Answer

4 with multiplicity 2 means one factor is $(x-4)^2$
then you have to find a quadratic with zeros $-5+3i$
do you know how to do that? it is not that hard

anonymous · Answer

turn the roots into factor form of the polynomial

anonymous · Answer

unfortunately Im not sure how to find the zeros, I've checked examples, but I havnt grasped how though :(

anonymous · Answer

hold the phone

anonymous · Answer

you are not being asked to find the zeros, you are TOLD the zeros
you have to come up with the polynomial

anonymous · Answer

you have "zeros -5+3i; 4 multiplicity 2 "

anonymous · Answer

sorry I meant find a quadractic with zeros

anonymous · Answer

"4 multiplicity 2" means it has two factors of $x-4$ i.e. it has a factor of $(x-4)^2$ 
here is a simple method to find the quadratic
work backwards, it requires a little algebra, then i can show you a method that requires only  memorizaton

anonymous · Answer

put 
$$x=-5+3i$$ work backwards by adding $5$ to get 
$$x+5=3i$$ then square both sides (carefully) and get 
$$x^2+10x+25=(3i)^2=-9$$ then add $9$ and get 
$$x^2+10x+34=0$$

anonymous · Answer

so the quadratic with zeros $-5+3i$ and $-5-3i$ is $x^2+10x+34$

anonymous · Answer

if you want a really really snap way, that requires no algebra but it does require memorizing, is that the quadratic with zeros $a+bi$ is 
$$x^2-2ax+(a^2+b^2)$$

anonymous · Answer

in your case you had $-5+3i$ so $a=-5,b=3$ and the quadratic is 
$$x^2-2\times(-5)x+(5^2+9^2)=x^2+10x+34$$

anonymous · Answer

ok I have a question, when you said square both sides (carefully), where did you get the 10x from?

anonymous · Answer

and your final job is to multiply out 
$$(x^2+10x+34)(x-4)^2$$ i would cheat so as not to screw up the algebra

anonymous · Answer

lol that is exactly what i meant by "carefully"

anonymous · Answer

$(x+5)^2=(x+5)(x+5)=x^2+10x+25$ and not for example $x^2+25$

anonymous · Answer

ohh lol ok I understand now

anonymous · Answer

Thank you so much satelite73!

anonymous · Answer

yw
you still have to multiply all that muck out "carefully"!

anonymous · Answer

haha oh I will!

anonymous · Answer

if you get this http://www.wolframalpha.com/input/?i=%28x^2%2B10x%2B34%29%28x-4%29^2 you know you are right

anonymous · Answer

Ok cool, i'll check it out then