Question

Find all rational zeros of the function f(x)=2x'-9x'-6x+5.

Answer 1

f(x)=2x^3-9x^2-6x+5

Factors of constant term P=5 are 1,-1,5,-5

Factors of leading coefficient q=2 are 1,-1,2,-2

Possible value of p/q which can be rational zero=1/1,1/-1,1/2,1/-2,-1/1,-1/-1,-1/2,-1/-2,5/1,5/-1,5/2,5/-2,-5/1,-5/-1,5/2,-5/-2

or, 1,-1,1/2,-1/2,5,-5,-5/2,5/2

We have to check these value for f(x) to see if is zero or not.

f(1)=2*1^3-9*1^2-6*1+5=-8 So, 1 is not the zero

f(-1)=2*(-1)^3-9*(-1)-6*(1)+5=0

Also

f(1/2)=0

f(-1/2)=5.5

f(5)=0

f(-5)=-440

f(-5/2)=-135/2

f(5/2)=70

So, f(x) is zero at x=-1,1/2,5

f(x) has rational zero at -1,1/2,5

Check

(x-(-1))(x-1/2)(x-5)=(x+1)(2x-1)(x-5)/2=(2x^3-9x^2-6x+5)/2

So, Answer is -1,1/2,5