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Question
If 3 and –3 are two zeroes of the polynomial `(x^4 + x^3 – 11x^2 – 9x + 18)`, find all the zeroes of the given polynomial.
Solution
Let `x^4 + x^3 – 11x^2 – 9x + 18`
Since 3 and – 3 are the zeroes of f(x), it follows that each one of (x + 3) and (x – 3) is a factor of f(x).
Consequently, `(x – 3) (x + 3) = (x^2 – 9)` is a factor of f(x).
On dividing `f(x) by (x^2 – 9)`, we get:
`f(x) = 0 ⇒ (x^2 + x – 2) (x^2 – 9) = 0`
⇒ `(x^2 + 2x – x – 2) (x – 3) (x + 3)`
⇒ `(x – 1) (x + 2) (x – 3) (x + 3) = 0`
⇒ `x = 1 or x = -2 or x = 3 or x = -3`
Hence, all the zeroes are 1, -2, 3 and -3.
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