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 May 28th, 2010, 11:27 AM #1 Newbie   Joined: May 2010 Posts: 3 Thanks: 0 Finding Polynomial with Roots of Real and Complex Numbers Find the polynomial of lowest degree with rational coefficients that has -2 and -3+5i as two of its roots.
May 28th, 2010, 05:51 PM   #2
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Re: Finding Polynomial with Roots of Real and Complex Numbers

Hello, HFH!

Quote:
 Find the polynomial of lowest degree with rational coefficients that has $-2$ and $-3+5i$ as two of its roots.

Complex roots occur in conjugate pairs.
Since $-3+5i$ is a root, then $-3-5i$ is also a root.

Hence:[color=beige] .[/color]$(x+2),\;\left(x-[-3+5i]\right),\;\left(x-[-3-5i]\right)\,$ are factors of the polynomial.

The simplest polynmial is:[color=beige] .[/color]$f(x) \;=\;(x\,+\,2)(x\,+\,3\,-\,5i)(x\,+\,3\,+\,5i)$

[color=beige]. . . . . . . . . . . . . . . . . . [/color]$f(x) \;=\;x^3\,+\,8x^2\,+\,46x\,+\,68$

 May 29th, 2010, 07:47 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,931 Thanks: 2205 Equations have roots; polynomials (and other functions) have zeros.
May 29th, 2010, 01:19 PM   #4
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Re:

Quote:
 Originally Posted by skipjack Equations have roots; polynomials (and other functions) have zeros.
You're quibbling.

 Tags complex, finding, numbers, polynomial, real, roots

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