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प्रश्न
Given that `x - sqrt(5)` is a factor of the cubic polynomial `x^3 - 3sqrt(5)x^2 + 13x - 3sqrt(5)`, find all the zeroes of the polynomial.
उत्तर
Let p(x) = `x^3 - 3sqrt(5)x^2 + 13x - 3sqrt(5)`
Using division algorithm,
`x^2 - 2sqrt(5)x + 3`
`(x - sqrt(5))")"overline(x^3 - 3sqrt(5)x^2 + 13x - 3sqrt(5)`
`x^3 - sqrt(5)x^2`
(–) (+)
`-2sqrt(5)x^2 + 13x - 3sqrt(5)`
`-2sqrt(5)x^2 + 10x`
(+) (–)
`3x - 3sqrt(5)`
`3x - 3sqrt(5)`
(–) (+)
0
∴ `x^3 - 3sqrt(5)x^2 + 13x - 3sqrt(5)`
= `(x^2 - 2sqrt(5)x + 3)(x - sqrt(5))`
= `(x - sqrt(5))[x^2 - {(sqrt(5) + sqrt(2)) + (sqrt(5) - sqrt(2))x + 3]`
= `(x - sqrt(5))[x^2 - (sqrt(5) + sqrt(2))x - (sqrt(5) - sqrt(2))x + (sqrt(5) + sqrt(2))(sqrt(5) - sqrt(2))]`
= `(x - sqrt(5)){x - (sqrt(5) + sqrt(2))){x - (sqrt(5) - sqrt(2))}`
So, all the zeroes of the given polynomial are`sqrt(5), (sqrt(5) + sqrt(2)) and (sqrt(5) - sqrt(2))`.
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