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Question
If 1 and –2 are two zeroes of the polynomial `(x^3 – 4x^2 – 7x + 10)`, find its third zero.
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Solution
`Let f(x) = 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 `(x^2 + x – 2)`, we get:
`f(x) = 0 ⇒ (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.
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