solving polynomial

**juldancer** · September 28th 2008, 07:28 PM

Find an integer d such that the equation x^3 + 4x^2 - 9x + d = 0 has two roots that are additive inverses of each other.

Does anyone know how to do this?

**Jhevon** · September 28th 2008, 07:42 PM

Originally Posted by juldancer

Find an integer d such that the equation x^3 + 4x^2 - 9x + d = 0 has two roots that are additive inverses of each other.

Does anyone know how to do this?

yup

first recall what additive inverses are. the additive inverse of a number $a$ is $-a$ .

also recall that a cubic where the coefficient of $x^3$ is 1 can be written as $(x - r_1)(x - r_2)(x - r_3) = 0$ where $r_1,r_2,$ and $r_3$ are the roots (not necessarily all real) of the equation.

so let $a$ and $-a$ be the two special roots. call the other root $b$ . thus we have

$(x - a)(x + a)(x - b) = 0$

$\Rightarrow x^3 - bx^2 - a^2 x + ab = 0$

now equate coefficients to get the values of $a$ and $b$ , and hence find $d$

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