Is 5/4 a root?

Question

Is 5/4 a root of 4x^7+6x^6+3x^5+250^4+1000x+75x^2-20x+17=0?

1000x^3 not just 1000x

Answer 1

Is 5/4 a root of 4x^7 +6x^6 +3x^5 +250x^4 +1000x^3 +75x^2 -20x +17 = 0 ?
Call the expression f(x).
It doesn't seem to have any obvious factorizations.

We want to test if f(5/4) = 0

f(5/4) = 4x^7 +6x^6 +3x^5 +250x^4 +1000x^3 +75x^2 -20x +17

Note for +ve x, there's only one negative term, the -20x. All other terms are positive.
So if suffices to show that f(x) > 1000x^3-20x +17 =
1000(1.25)^3 -20(1.25) +17
which is > 1000 - 25+17 = 992
which is clearly > 0

Thus 5/4 is not a root.


[You can graph it with MatLab, NumPython or this rootfinder:
http://comp.uark.edu/~icjong/docu/software/digital80/digital80.html with
Y = 4*pow(X,7) +6*pow(X,6) +3*pow(X,5) +250*pow(X,4) +1000*pow(X,3) +75*pow(X,2) -20*X +17

Answer 2

Actually, there's a much simpler method than smci's -- simply note that by the rational root theorem, any rational root of this polynomial would have to take the form p/q, where p|17 and q|4. So the only possible rational roots of this polynomial are ±1, ±1/2, ±1/4, ±17, ±17/2, or ±17/4. Since 5/4 is not one of these numbers, it cannot possibly be a root of the indicated polynomial.