Find all the zeros of the polynomial x4 + x3 − 34x2 − 4x + 120,


Find all the zeros of the polynomial $x^{4}+x^{3}-34 x^{2}-4 x+120$, if two of its zeros are 2 and $-2$.


We know that if $x=\alpha$ is a zero of a polynomial, then $x-\alpha$ is a factor of $f(x)$.

Since, 2 and $-2$ are zeros of $f(x)$.




$x^{2}-4$ is a factor of $f(x)$.Now, we divide $x^{4}+x^{3}-34 x^{2}-4 x+120$ by $g(x)=x^{2}-4$ to find the other zeros of $f(x)$.

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

$x^{4}+x^{3}-34 x^{2}-4 x+120=\left(x^{2}-4\right)\left(x^{2}+x-30\right)-0$

$x^{4}+x^{3}-34 x^{2}-4 x+120=(x+2)(x-2)\left(x^{2}+6 x-5 x-30\right)$

$x^{4}+x^{3}-34 x^{2}-4 x+120=(x+2)(x-2)(x(x+6)-5(x+6))$

$x^{4}+x^{3}-34 x^{2}-4 x+120=(x+2)(x-2)(x+6)(x-5)$

Hence, the zeros of the given polynomial are $-2,+2,-6$, and 5

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