Write a polynomial function with rational coefficients so that P(x)=0 has the given roots. #16. 2 + i and 1-√5 #15. 3i and √6

Question

anonymous · Answer

ick

anonymous · Answer

i take it these are two separate problems right?

anonymous · Answer

Yea

anonymous · Answer

which one you want to do first?
#15 is the easier of the two

anonymous · Answer

Lets tackle that one first then.

anonymous · Answer

ok

anonymous · Answer

the conjugate of $3i$ is $-3i$ and the quadratic with those two roots is 
$$(x-3i)(x+3i)=x^2+9$$ quick and easy

anonymous · Answer

the conjugate of $\sqrt6$ is $-\sqrt6$ and the quadratic with those zeros is 
$$(x-\sqrt6)(x+\sqrt6)=x^2-6$$

anonymous · Answer

your job is now to multiply out 
$$(x^2+9)(x^2-6)$$ without making an algebra mistake
you good for that one?

anonymous · Answer

Eh..I'm stuck on multiplying the squares

anonymous · Answer

$x^2\times x^2=x^4$

anonymous · Answer

you get $$x^4-6x^2+9x^2-54$$ or 
$$x^4+3x^2-54$$

anonymous · Answer

ok for that one?
the second one is harder

anonymous · Answer

ya

anonymous · Answer

you need a quadratic with $2+i$ as a zero

anonymous · Answer

there are several ways to find it
i will show you an easy one, then a really really easy one
but the really really easy one requires memorizing something

anonymous · Answer

Thats fine

anonymous · Answer

work backwards 
$$x=2+i\
x-2=i$$ then square both sides and get $$(x-2)^2=i^2$$ or 
$$x^2-4x+4=-1$$ add $1$ and get 
$$x^2-4x+5$$ as your quadratic

anonymous · Answer

the really really easy way is to know that if $a+bi$ is a zero, then the quadratic is 
$$x^2-2ax+(a^2+b^2)$$ in your case you have $2+i$ so $a=2,b=1$ and the quadratic is 
$$x^2-2\times 2x+(2^2+1^2)=x^2-4x+5$$

anonymous · Answer

as for $1+\sqrt5$ without the $i$ it is 
$$x^2-2x+(1^2-\sqrt5)^2=x^2-2x-4$$

anonymous · Answer

your last job is to multiply out 
$$(x^2-4x+5)(x^2-2x-4)$$ i would cheat

anonymous · Answer

Okay here's another thing I'm having trouble w/ ... Finding all rational roots for P(x)=0 
#26. P(x)=x^3-x^2-34x-56

anonymous · Answer

cheat

anonymous · Answer

does it say "find the rational roots" or "find all POSSIBLE rational roots"?

anonymous · Answer

here is the solution http://www.wolframalpha.com/input/?i=x^3-x^2-34x-56

anonymous · Answer

It just says "Find all rational roots"

anonymous · Answer

open the link, there they are

anonymous · Answer

It doesn't show me how it found the roots. Can you show me how to do one so I can do the rest?

anonymous · Answer

P(x)=3x^3+x^2-8x-12