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

Given, 

f(x)= 2x⁴ + x³ – 14x² – 19x – 6 

If the two zeros of the polynomial are -2 and -1, then its factors are (x + 2) and (x + 1) 

⇒ (x+2)(x+1) = x² + x + 2x + 2 

= x² + 3x +2 …… (i) 

This means that (i) is a factor of f(x). So, performing division algorithm we get,

The quotient is 2x² – 5x – 3. 

⇒ f(x)= (2x² – 5x – 3)( x² + 3x +2) 

For obtaining the other 2 zeros of the polynomial 

We put, 

2x² – 5x – 3 = 0

⇒ (2x + 1)(x – 3) = 0 

∴ x = -1/2 or 3 

Hence, all the zeros of the polynomial are -2, -1, -1/2 and 3.