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factor x^2 - 30*x + 2817 in sqrt(-2)

asked 2016-11-02 00:47:32 -0500

Sha gravatar image

Is there a way I can use SAGE to factor my polynomial x^2 - 30*x + 2817 in sqrt(-2).

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Note that we write Sage rather than SAGE.

slelievre gravatar imageslelievre ( 2016-11-02 10:56:35 -0500 )edit

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answered 2016-11-02 01:22:36 -0500

rtc gravatar image

updated 2016-11-02 01:46:04 -0500

I think you are looking for the roots of the polynomial:

f = x^2 - 30*x + 2817
f.roots()

which gives:

[(-36*I*sqrt(2) + 15, 1), (36*I*sqrt(2) + 15, 1)]

This would mean that your original function is equal to:

$$ x^2-30x+2817 = \left(x-(15-36 \sqrt{-2})\right)\left(x-(15+36\sqrt{-2})\right)$$

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yes this is what I am looking for.. thank you..

Sha gravatar imageSha ( 2016-11-02 03:07:24 -0500 )edit
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answered 2016-11-03 05:32:30 -0500

Note that you can also work over the field QQbar of algebraic numbers.

Here is how you would factor your polynomial there and find its roots.

sage: P.<x> = QQbar[]
sage: p = x^2 - 30*x + 2817
sage: p.factor()
(x - 15.00000000000000? - 50.91168824543143?*I) * (x - 15.00000000000000? + 50.91168824543143?*I)
sage: p.roots()
[(15.00000000000000? - 50.91168824543143?*I, 1),
 (15.00000000000000? + 50.91168824543143?*I, 1)]

If you want a radical expression for the roots:

sage: for r in p.roots():
....:     print(r[0].radical_expression())
....:     
-36*I*sqrt(2) + 15
36*I*sqrt(2) + 15
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Thank you for explaining this to me. It is so useful in my calculation.

Sha gravatar imageSha ( 2016-11-03 05:50:38 -0500 )edit

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Asked: 2016-11-02 00:47:32 -0500

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Last updated: Nov 03 '16