Question

Obtain all zeroes of the polynomial f(x)=2x⁴+x³-14x²-19x-6 if two of its zeroes are -2 and -1

Answer 1

f(x)= 2x^4+x^3- 14x^2- 19x-6
Zeroes = -2 , -1

alpha = -2 , beta = -1
Now ,first do sum and then product of zeroes .
sum = alpha +beta = -2-1= -3
product = alpha × beta = -2×-1=2

Now form the polynomial.
Polynomial = k ( x^2 -(sum ) x + product )
= k ( x^2 -(-3)x + 2 )
= k( x^2 +3x +2)
now take value of k as 1 ( in all cases )
for k= 1 , the polynomial = x^2 + 3x +2
= g( x) = x^2+ 3x +2

You have got the g( x ) . You can divide p( x) with g( x) .