Thread: roots of polynomial equations (complex #s)

so im just starting to learn about roots of polynomial equations with roots that are complex numbers and i dont understand them at all.. i hope you can help

A. Use the quadratic formula to find the roots of the equation x^2+4x+5=0. Simplify and compare the roots. what do you notice?
B. write a quadratic equation with integral coefficients such that one of its roots is 4-5i
C. Write a quartic equation with integral coefficients and with roots 7i and -3i
.....
so for A i think its x= -2+i and x=-2-i but i dont "notice" anything and have no idea how to do the other questions.
im thinking its something simple and im going to feel stupid once i find out the answer -.-

The roots are conjugate pairs

Originally Posted by phthiriasis

so im just starting to learn about roots of polynomial equations with roots that are complex numbers and i dont understand them at all.. i hope you can help

A. Use the quadratic formula to find the roots of the equation x^2+4x+5=0. Simplify and compare the roots. what do you notice?
B. write a quadratic equation with integral coefficients such that one of its roots is 4-5i
C. Write a quartic equation with integral coefficients and with roots 7i and -3i
.....
so for A i get x= -2+i and x=-2-i but i dont "notice" anything and have no idea how to do the other questions.
im thinking its something simple and im going to feel stupid once i find out the answer -.-

What they wanted you to notice is that

are conjugates two complex numbers are conjugate if

the numbers are the same, but the immaginary parts have opposite signs.

Now with this new knowlege

we can find the roots of the other quadratic.

since we know that 4-5i is a root then 4+5i must also be a root.

If c is a root of a polynomial then x-c is a facor of the polynomial

now comes the fun part

We multiply out our two factors.





I hope this helps
see what you can do with the next one.

thanks a lot TheEmptySet

so i tried the next one..
since 7i and -3i are roots the other roots are -7i and 3i
so i multiplied it all out..
(x-7i)(x+7i)(x-3i)(x+3i) and ended up with
x^4+58x^2+441
is this right?^^
thanks again!