Obtain all other zeros of (x4 + 4x3 ‒ 2x2 ‒ 20x ‒15) if two of its zeros are √5 and – √5.
Let us assume f (x) = x4 + 4x3 ‒ 2x2 ‒ 20x - 15
As (x-√5) and (x-√5) are the zeros of the given polynomial therefore each one of (x-√5) and (x + √5) is a factor of f (x)
Consequently, (x-√5) (x + √5) = (x2 – 5) is a factor of f (x)
Now, on dividing f (x) by (x2 – 5) we get:
f (x) = 0
x4 + 4x3 – 7x2 – 20x – 15 = 0
(x2 – 5) (x2 + 4x + 3) = 0
(x -√5) (x + √5) (x + 1) (x + 3) = 0
∴ x = √5 or x = - √5 or x = - 1 or x = - 3
Hence, all the zeros of the given polynomial are √5, -√5, - 1 and - 3