# Find the value of k if f(x) has three distinct real roots ?

## ${x}^{3} - 3 x + k = 0$

Jun 26, 2018

See explanation below

#### Explanation:

If equation has three real roots all of them are distinct, then

$f \left(x\right) = \left(x - a\right) \left(x - b\right) \left(x - c\right)$ where $a , b , c$ are roots of $f \left(x\right)$

Developing:

$f \left(x\right) = \left({x}^{2} - b x - a x + a b\right) \left(x - c\right) = {x}^{3} - c {x}^{2} - b {x}^{2} + b c x - a {x}^{2} + a c x + a b x - a b c = {x}^{3} - \left(a + b + c\right) {x}^{2} + \left(a b + a c + b c\right) x - a b c$

We know that two polynomial expresions are equal if and only if his coefficients are equal. Then (Cardano formulae)

$a + b + c = 0$
$a b + a c + b c = - 3$
$- a b c = k$

Now, if we take a look to ${x}^{3} - 3 x$ we see that has 3 roots if we re-write in form $x \left({x}^{2} - 3\right)$ ; $x = 0 , x = \pm \sqrt{3}$

Adding +k we are traslating graph up or down if k is positive or negative respectively and this fact we can eliminate 2 real roots adding 2 complex roots.(if a polinomial has a complex root, then his conjugate is also root). Hope this helps

Jun 26, 2018

$- 2 \setminus \le k \setminus \le 2$

#### Explanation:

Let's start by taking a look at ${x}^{3} - 3 x$: it has three roots, and local minimum and a maximum:

graph{x^3-3x [-2 2 -3 3]}

The minimum and maximum are found by posing

$f ' \left(x\right) = 3 {x}^{2} - 3 = 0 \setminus \iff {x}^{2} - 1 = 0 \setminus \iff x = \setminus \pm 1$

With corresponding values $f \left(1\right) = 1 - 3 = - 2$ for the minimum and $f \left(- 1\right) = - 1 + 3 = 2$ for the maximum.

Adding $k$ to the function translates the graph upwards ($k > 0$) or downwards ($k < 0$). You can see that the function has three zeroes as long as the maximum is above the $x$ axis, and the minimum is below.

If we translate the function up three units, for example, the minimum would be above the $x$ axis, and we would lose solutions:

graph{x^3-3x+3 [-5 5 -1 6]}

Similarly, we can't translate the graph down more than two units: see this example with $k = - 3$

graph{x^3-3x-3 [-5 5 -6 1]}

While any $k$ between $- 2$ and $2$ is ok:

graph{x^3-3x+1 [-3 3 -2 4]}