Given polynomial $f(x)=x^3+2x^2-51x+108$ and a factor $x+9$ how do you find all other factors?

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Answer 1 · 2016-12-31T16:20:08

The answer is $=(x+9)(x-4)(x-3)$

$f(x)=x^3+2x^2-51x+108$

$(x+9)$ is a factor

We do a long division

$color(white)(aaaa)$ $x^3+2x^2-51x+108$ $color(white)(aaaa)$ $∣$ $x+9$

$color(white)(aaaa)$ $x^3+9x^2$ $color(white)(aaaaaaaaaaaaaaa)$ $∣$ $x^2-7x+12$

$color(white)(aaaa)$ $0-7x^2-51x$

$color(white)(aaaaaa)$ $-7x^2-63x$

$color(white)(aaaaaaaa)$ $-0+12x+108$

$color(white)(aaaaaaaaaaaa)$ $+12x+108$

$color(white)(aaaaaaaaaaaaaa)$ $+0+0$

Therefore,

$(x^3+2x^2-51x+108)/(x+9)=x^2-7x+12$

We can factorise the quotient

$x^2-7x+12=(x-4)(x-3)$

Given polynomial f(x)=x^3+2x^2-51x+108f(x)=x^3+2x^2-51x+108 and a factor x+9x+9 how do you find all other factors?