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Show that Discriminant $D$ of

$x^4+7x^3+14x^2+3x-9$

is $0$ WITHOUT using the discriminant formula. If this is true, what root properties does this specific polynomial hold and do the factors (for a certain $x$ value evaluated at this polynomial) have a specific form?

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    Just remember the *definition* of the discriminant, don't worry about the formula. – 2017-02-16

1 Answers 1

1

If the discriminant is zero, there are multiple roots (have a look here).

By inspection $x=-3$ is a root of equation $x^4+7x^3+14x^2+3x-9=0$. Using long division $$x^4+7x^3+14x^2+3x-9=(x+3)(x^3+4 x^2+2 x-3)$$ By inspection $x=-3$ is a root of equation $x^3+4 x^2+2 x-3=0$. Continuing with long division$$x^4+7x^3+14x^2+3x-9=(x+3)^2(x^2+x-1)$$