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#1**+4 **

x^3 + 2x^2 - 7x - 2

We have something called the Rational Roots Theorem here that might help us....without going into detail ...the * possible * rational roots are 1, - 1 , 2 or -2

I see that 1 is not a root

I see that -1 isn't either

But 2 is a root because (2)^3 + 2(2)^2 -7(2) - 2 = 8 + 8 - 14 - 2 = 0

So....using synthetic division, we can find the remaining polynomial

2 [ 1 2 - 7 -2 ]

2 8 2

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1 4 1 0

The remaining polynomial is x^2 + 4x + 1

We can complete the square on x to find the other two roots

x^2 + 4x + 4 = - 1 + 4

(x + 2)^2 = 3 take both roots

x + 2 = ±√ 3 subtract 2 from both sides and we get the other two solutions

x = √3 - 2 or x = -√3 - 2

CPhill Feb 19, 2019