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Question
Find all the zeros of the polynomial x3 + 3x2 − 2x − 6, if two of its zeros are `-sqrt2` and `sqrt2`
Solution
We know that if x = a is a zero of a polynomial, then x - a is a factor of f(x).
Since, `sqrt2` and `-sqrt2` are zeros of f(x).
Therefore
`(x+sqrt2)(x-sqrt2)=x^2-(sqrt2)^2`
= x2 - 2
x2 - 2 is a factor of f(x). Now, we divide x3 + 3x2 − 2x − 6 by g(x) = x2 - 2 to find the zero of f(x).
By using division algorithm we have
f(x) = g(x) x q(x) - r(x)
x3 + 3x2 − 2x − 6 = (x2 - 2)(x + 3) - 0
x3 + 3x2 − 2x − 6 `=(x+sqrt2)(x-sqrt2)(x+3)`
Hence, the zeros of the given polynomials are `-sqrt2`, `+sqrt2` and -3.
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