find the root of $\mathbb{R} $:
$$x^8-24x^4+256x^2+144=0$$
my try:
$$t=x^2\\t^4-24t^2+256t+144=0\\t^4-(8×3)t^2+(8×2^5)t+(8×18)=0$$
now ?!!
thank you very much!
find the root : $x^8-24x^4+256x^2+144=0$
3
$\begingroup$
polynomials
roots
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1To find answer for such questions, use www.wolframalpha.com – 2017-01-23
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0It is a good start. The next thing I would check is Rational Roots test, which says (since your polynomial in $t$ is monic) that any rational root must be an integer that divides $144$. This is an important theme in algebra, so it's worth checking these in order to better appreciate this test. – 2017-01-23
2 Answers
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Note that $$x^8-24x^4+144=(x^4-12)^2$$
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The equation can be factored as $$(x^2-8x+36)(x^2+8x+4)=0$$ So here are the roots: $$x_1 = -2(2+\sqrt{3})$$ $$x_2 = -2(2-\sqrt{3})$$ $$x_3 = 4-2\sqrt{5}i$$ $$x_4 = 4+2\sqrt{5}i$$
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2So how do you arrive at those factors? – 2017-01-23