g(x)=x^3-4x^2+x-10

Question

lit the rational zeros

Answer 1

g(0)=-10. The even powers of x have negative coefficients and the odd powers of x have positive coefficients, therefore when x<0 g(x)<-10. So a rational, non-complex zero can’t be less than zero.

g(1)=1-4+1-10=-12, g(2)=8-16+2-10=-16, g(3)=27-36+3-10=-16, g(4)=64-64+4-10=-6, g(5)=125-100+5-10=20.

The sign changes between g(4) and g(5) so there must be a zero when 4<x<5.

We can then investigate the interval (4,5) closer, by finding g(4.1), g(4.2), ..., g(4.9).

If we do this, 4.3<x<4.4. Now we look more closely at (4.3,4.4).

We get 4.30<x<4.31. And so on.