Question

Show that 3 is a zero of the polynomial 2x 3 - x 2 - 13x - 6. hence find all the other zeroes of this polynomial and verify the relation between zeros and coefficients

Answer 1

2(3)^3-(3)^2-13*3-6=0
2*27-9-39-6=0
54-54=0
0=0
x=3
x-3=0
2x^3-x^2-13x-6÷x-3
=2x^2+5x+2
by splitting middle term
2x^2+4x+x+2=0
2x(x+2)+1(x+2)=0
(2x+1)(x+2)=0
either           or
2x+1=0        x+2=0
2x=-1           x=-2
x=-1/2           
here,a=2 b=5 c=2  α=-1/2 β=-2
α+β=-b/a                       αβ=c/a
-1/2+(-2)=-5/2               (-1/2)(-2)=2/2
-1/2-4/2=-5/2                 2/2=2/2
-5/2=-5/2                         1=1

hence verified