if the two of its zeroes are −√3 and √3,
obtain all the zeroes of the polynomial.
Class 10th board, polynomials 3rd exercise.
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Obtain all zeroes of the polynomial f(x) = x4 – 3×3 – x2 + 9x – 6, if the two of its zeroes are −√3 and √3.
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Solution:
Given,
f(x) = x4 – 3x3 – x2 + 9x – 6
Since two of the zeroes of the polynomial are −√3 and √3 so, (x + √3) and (x–√3) are factors of f(x).
⇒ x2 – 3 is a factor of f(x). Hence, performing division algorithm, we get
⇒ f(x)= (x2 – 3x + 2)( x2 – 3)
So, by putting x2 – 3x + 2 = 0, we can get the other 2 zeros.
⇒ (x – 2)(x – 1) = 0
∴ x = 2 or 1
Hence, all the zeros of the polynomial are −√3, 1, √3 and 2.