f(x) = x^4 + x^3 - 3x^2 - 7x - 4 Find this sroot and then use the calculator to determine any other irrational roots (rounded to 3 decimals).  Enter answers smallest to largest.

Expert Answers
embizze eNotes educator| Certified Educator

Given `f(x)=x^4+x^3-3x^2-7x-4` find the real roots:

The only possible rational roots are `+-1,+-2,+-4` . Using synthetic division we find -1 is a root:

-1  |  1   1   -3   -7   -4
            -1    0     3    4
         1   0   -3    -4   0

So `x^4+x^3-3x^2-7x-4=(x+1)(x^3-3x-4)`

The second factor is a cubic with a positive leading coefficient. Since it is a cubic it has 1 real root or 3 real roots. As x tends to negative infinity the function decreases without bound and as x tends to positive infinity the function increases without bound. Since it is a cubic, it can have at most 2 turning points (points where the function changes from increasing to decreasing or decreasing to increasing.)

The graph of the second factor:

There is only 1 real root -- using a graphing utility the estimate for the root is `x~~2.1958333`


The real roots for `f(x)=x^4+x^3-3x^2-7x-4` are x=-1 and `x~~2.196`


The graph of f(x):