SOLUTION: The complex number 2+i is a root of {{{z^3-z^2-pz+15=0}}}, where p is real. Explain why 2-i is another root and find the third root.

Question 295818: The complex number 2+i is a root of , where p is real. Explain why 2-i is another root and find the third root.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Because complex roots ALWAYS come in conjugate pairs. Therefore if

is a root of a polynomial equation, then

is also a root of that same polynomial equation. Guaranteed.

If

is the root of a polynomial equation, then

is a factor of the polynomial.

Since you know two of the roots of the equation, you know two of the factors of the polynomial, namely:

and

So, step one is to multiply these two factors together. I'll leave that as an exercise for the student. Hint: The product of two conjugates is the difference of two squares. And don't forget

The next step is to divide the original polynomial by the quadratic trinomial you just derived in the previous step. Use polynomial long division. See http://www.purplemath.com/modules/polydiv2.htm if you need a refresher on the process. You will need to select an appropriate value for

so that you end up with a zero remainder. The quotient you obtain will be the third factor.

Good luck.

John

RELATED QUESTIONS
The complex number 2+i is a root of z^3-z^2-pz+15=0, where p is real. Explain why 2-i is... (answered by stanbon,richwmiller)

one solution of z3+ A z^2 -z-14 =0 is z=-2-root 3 i where A is a real number. find A and (answered by math_tutor2020,ikleyn)

Can someone please help me with this question - Thanks Let P(z)= {{{ z^3-kz^2+22z-20 }}} (answered by KMST)

given that the equation Z^2 + (p+iq)Z + r + is = 0 where p,q,r,s are real and non-zero... (answered by richard1234)

z is the complex number 1+i, where i^2=-1. (i) Find z^2 and z^3. (ii) Verify that... (answered by ewatrrr)

sir, i am doing B.Sc(maths) 1st year at madurai kamaraj university through distance... (answered by stanbon)

Check that (2+i) is a root of z^4+2(z^3)-9(z^2)-10z+50=0. Find the remaining three... (answered by MathLover1,MathTherapy)

Find the number of times r is a root of P(x)=0. P(x)= x^4+4x^3-16x-16; r=-2 i found... (answered by Edwin McCravy,jim_thompson5910,solver91311)

Write the complex number z = 2 + 6i in polar form: z = r(cos theta + i sin theta) where... (answered by Alan3354)