How do you find all rational zeroes of the function using synthetic division $f (x) = x^{4} + x^{3} + x^{2} - 9 x - 10$ $f(x)=x^4+x^3+x^2-9x-10$ ?

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Answer 1 · 2018-01-14T16:34:53

$x_{1} = - 1 - 2 i$ $x_1=-1-2i$ , $x_{2} = - 1 + 2 i$ $x_2=-1+2i$ , $x_{3} = - 1$ $x_3=-1$ and $x_{4} = 2$ $x_4=2$

After using Rational Roots Test, $x = - 1$ $x=-1$ is a root of the polynomial. Hence $x + 1$ $x+1$ is multiplier of it. Consequently,

$x^{4} + x^{3} + x^{2} - 9 x - 10$ $x^4+x^3+x^2-9x-10$

= $x^{3} \cdot (x + 1) + (x + 1) \cdot (x - 10)$ $x^3*(x+1)+(x+1)*(x-10)$

= $(x + 1) \cdot (x^{3} + x - 10)$ $(x+1)*(x^3+x-10)$

= $(x + 1) \cdot (x^{3} - 8 + x - 2)$ $(x+1)*(x^3-8+x-2)$

= $(x + 1) [(x - 2) \cdot (x^{2} + 2 x + 4) + x - 2]$ $(x+1)[(x-2)*(x^2+2x+4)+x-2]$

= $(x + 1) (x - 2) (x^{2} + 2 x + 5)$ $(x+1)(x-2)(x^2+2x+5)$

Hence roots of $f (x)$ $f(x)$ are $- 1, 2, - 1 + 2 i$ $-1, 2, -1+2i$ and $- 1 - 2 i$ $-1-2i$

How do you find all rational zeroes of the function using synthetic division f(x)=x4+x3+x2−9x−10f(x)=x^4+x^3+x^2-9x-10?