Find the value of k if f(x) has three distinct real roots ?
#x^3-3x+k=0#
2 Answers
See explanation below
Explanation:
If equation has three real roots all of them are distinct, then
Developing:
We know that two polynomial expresions are equal if and only if his coefficients are equal. Then (Cardano formulae)
Now, if we take a look to
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
Explanation:
Let's start by taking a look at
graph{x^3-3x [-2 2 -3 3]}
The minimum and maximum are found by posing
With corresponding values
Adding
If we translate the function up three units, for example, the minimum would be above the
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
graph{x^3-3x-3 [-5 5 -6 1]}
While any
graph{x^3-3x+1 [-3 3 -2 4]}