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Question
Obtain all other zeroes of `(x^4 + 4x^3 – 2x^2 – 20x – 15)` if two of its zeroes are `sqrt5 and –sqrt5.`
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Solution
The given polynomial is` f(x) = x^4 + 4x^3 – 2x^2 – 20x – 15.`
Since `(x – sqrt5) and (x + sqrt5)` are the zeroes of f(x) it follows that each one of `(x – sqrt5) and (x + sqrt5)` is a factor of f(x).
Consequently, `(x – sqrt5) (x + sqrt5) = (x2 – 5)` is a factor of f(x).
On dividing f(x) by (x2 – 5), we get:
`f(x) = 0`
`⇒ x^4 + 4x^3 – 7x^2 – 20x – 15 = 0`
`⇒ (x^2 – 5) (x2 + 4x + 3) = 0`
`⇒ (x – sqrt5) (x + sqrt5) (x + 1) (x + 3) = 0`
`⇒ x = sqrt5 or x = -sqrt5 or x = -1 or x = -3`
Hence, all the zeroes are sqrt5, -sqrt5, -1 and -3.
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