For any polynomial of form anxn + an−1xn−1 + ··· + a2x2 + a1x + a0 = 0:
Sum of roots
Product of roots:
Example: Given x2 + 8x + k = 0, find both roots, when one is 3 times larger than another. Then find value of k.
First we need to find out what both roots are. Let x1 = α, then x2 = 3α. Using formula we get:
Sum of roots: − 8/1
⇒ Thus: x1 + x2 = −8
α + 3α = −8
α = −2 = x1
x2 = 3 × −2 = −6
It means that we can factorise original polynomial as (x + 2)(x + 6) = 0, giving:
(x + 2)(x + 6) = x2 + 8x + 12
Therefore, k = 12.