SOLUTION: Please factor completely and show a detailed solution. Thank you! x^4 - 23x^2 + 49

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Question 1039621: Please factor completely and show a detailed solution. Thank you!
x^4 - 23x^2 + 49
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
p=x%5E2

p%5E2-23p%2B49

That method seems not useful.

Try Rational Roots Theorem and synthetic division to check roots -1, -7, 1, 7.

No remainders found (synthetic divisions done on paper).
The polynomial does not have rational roots. Binomial factors are not found.
It does have four real but irrational roots. Back to the substitution for x%5E2, you can find them.


-
p=x%5E2
p%5E2-23p%2B49
p=%2823%2B-+3%2Asqrt%2837%29%29%2F2

Substitute back the x%5E2.

x=0%2B-+sqrt%28%2823%2B-+3%2Asqrt%2837%29%29%2F2%29-----which is FOUR values; the 0 included so that the rendering on the page will work. I have not finished putting this into the final desired factored form.