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Question
If 2 and -2 are two zeroes of the polynomial `(x^4 + x^3 – 34x^2 – 4x + 120)`, find all the zeroes of the given polynomial.
Solution
Let f(x) = x^4 + x^3 – 34x^2 – 4x + 120
Since 2 and -2 are the zeroes of f(x), it follows that each one of (x – 2) and (x + 2) is a factor of f(x).
Consequently, (x – 2) (x + 2) = (x2 – 4) is a factor of f(x).
On dividing f(x) by `(x^2 – 4)`, we get:
f(x) = 0
⇒` (x2 + x – 30) (x^2 – 4) = 0`
⇒` (x^2 + 6x – 5x – 30) (x – 2) (x + 2)`
⇒ `[x(x + 6) – 5(x + 6)] (x – 2) (x + 2)`
⇒`(x – 5) (x + 6) (x – 2) (x + 2) = 0`
⇒` x = 5 or x = -6 or x = 2 or x = -2`
Hence, all the zeroes are 2, -2, 5 and -6.
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