Question

If 1 and –2 are two zeroes of the polynomial ${\displaystyle (x^3–4x^2–7x+10)}$, find its third zero.

Answer 1

${\displaystyle Letf\left(x\right)=x^3–4x^2–7x+10}$ 
Since 1 and –2 are the zeroes of f(x), it follows that each one of (x–1) and
(x+2) is a factor of f(x).
Consequently, (x–1) (x+2) = (x^2 + x – 2) is a factor of f(x).
On dividing f(x) by ${\displaystyle (x^2+x–2)}$, we get: 

 

${\displaystyle f\left(x\right)=0\Rightarrow (x^2+x–2)(x–5)=0}$
⇒ (x – 1) (x + 2) (x – 5) = 0
⇒ x = 1 or x = – 2 or x = 5
Hence, the third zero is 5.