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Obtain all the zeroes of the polynomial 2x^{4} − 5x^{3} − 11x^{2 }+ 20x + 12 when 2 and − 2 are two zeroes of the above polynomial.

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#### Solution

We know that if x = α is a zero of a polynomial, and then x - α is a factor of f(x).

Since 2 and −2 are zeros of f(x).

Therefore

(x − 2)(x + 2) = x^{2}−4

(x^{2 }− 4) is a factor of f(x).

Now, we divide 2x^{4} − 5x^{3 }− 11x^{2 }+ 20x +12 by g(x) = (x^{2 }− 4) to find the zero f(x).

By using division algorithm we have f(x) = g(x) x q(x) - r(x)

2x^{4 }− 5x^{3 }− 11x^{2 }+ 20x + 12 = (x^{2 }− 4)(2x^{2 }− 5x − 3)

=(x − 2)(x + 2)[2x(x − 3) + 1(x − 3)]

=(x − 2)(x + 2)(x − 3)(2x + 1)

Hence, the zeros of the given polynomial are 2, −2, 3, `−1/2`.

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