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प्रश्न
Find all the zeros of the polynomial x4 + x3 − 34x2 − 4x + 120, if two of its zeros are 2 and −2.
उत्तर
We know that if x = a is a zero of a polynomial, then x - a is a factor of f(x).
Since, 2 and -2 are zeros of f(x).
Therefore
(x + 2)(x - 2) = x2 - 22
= x2 - 4
x2 - 4 is a factor of f(x). Now, we divide x4 + x3 − 34x2 − 4x + 120 by g(x) = x2 - 4 to find the other zeros of f(x).
By using that division algorithm we have,
f(x) = g(x) x q(x) - r(x)
x4 + x3 − 34x2 − 4x + 120 = (x2 - 4)(x2 + x - 30) - 0
x4 + x3 − 34x2 − 4x + 120 = (x + 2)(x - 2)(x2 + 6x - 5x - 30)
x4 + x3 − 34x2 − 4x + 120 = (x + 2)(x - 2)(x(x + 6) - 5(x + 6))
x4 + x3 − 34x2 − 4x + 120 = (x + 2)(x - 2)(x + 6)
(x - 5)
Hence, the zeros of the given polynomial are -2, +2, -6 and 5.
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