Please help solve   x^4-x^3-10x^2+2x+4=0

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lfryerda | High School Teacher | (Level 2) Educator

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To solve this equation, we need to use the factor theorem to find the factors of the polynomial `f(x)=x^4-x^3-10x^2+2x+4`.

If the factors of the polynomial are integers, then they will also be factors of the constant term, which in this case is 4.  We guess possible factors and see which guesses make `f(x)=0` .

The problem is that guessing each of `+-1` , `+-2` , and `+-4` doesn't get any roots.

The next step with quartics is to try and factor the polynomial into the form `(x^2+px+q)(x^2+rx+s)` .  After expanding this factoring and comparing with the coefficients of the function, we see that this is

`f(x)=(x^2-3x-2)(x^2+2x-2)`

Setting each of these factors to zero and using the quadratic formula then gives the roots:

`x={3+-sqrt{9+4(2)}}/2`

`={3+-sqrt17}/2`

and

`x={-2+-sqrt{4+4(2)}}/2`

`=-1+-sqrt3`

The solutions to the equation are `x={3+-sqrt17}/2`

and `x=-1+-sqrt3` .

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