Draw the graph of `y=x^(4)+2x^(2)-8x+3`
Find the number of real roots of the equation `x^(4)+2x^(2)-8x+3=0`.
Also find the sum of the integral parts of all real roots.

Explanation

We have `f(x) =x^(4)+2x^(2)-8x+3`
`f(0)=3`
`f'(x)=4x^(3)+4x-8=4(x-1)(x^(2)+2+2)`
`f'(x)=0 :. x=1`
Hence the graph of `y=f(x)` has only one turning point, which is the point of minima.
When `xrarr+-oo,f(x)rarroo` as the leading coefficient is '1'
`f(1)=1+2-8+3 lt 0`
Hence the graph of `y=f(x)` can be drawn as shown in the following figure.

From the graph, `f(x)=0` has only two real roots.
`f(2)=(2)^(4)+2(2)^(2)-8(2)+3 gt 0`
Thus one lies in (0,1) and the other lies in (1, 2).
So the sum of the integral parts of real roots is 0+1=1.