Question

What is the general formula for the discriminant of a polynomial of degree \[n\] ?

Answer 1

Hint: In the question, we are asked to write the general formula for the discriminant of a polynomial of degree \[n\]. We will use the Sylvester matrix which involves the use of \[f(x)\] and \[f'(x)\]. We will then plot the matrix form of the Sylvester matrix and for \[n\] degree of a polynomial, the order of the matrix formed will be \[\left( 2n-1 \right)\times \left( 2n-1 \right)\]. Accordingly, we will then carry out the calculation based on the general formula for the discriminant of a polynomial.

Complete step-by-step solution:
According to the given question, we are asked to write the general formula for the discriminant of a polynomial of degree \[n\].
We will use here the Sylvester matrix to write the general formula for the discriminant of a polynomial. This will include the use of \[f(x)\] and \[f'(x)\].
Let \[f(x)\] be \[f(x)={{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+...+{{a}_{1}}x+{{a}_{0}}\]
Then, \[f'(x)\] will be,
\[f'(x)=n{{a}_{n-1}}{{x}^{n-1}}+\left( n-1 \right){{a}_{n-2}}{{x}^{n-2}}+...+{{a}_{1}}\]
The Sylvester matrix for a polynomial of degree \[n\] is formed having the order of the matrix as \[\left( 2n-1 \right)\times \left( 2n-1 \right)\]. And the matrix comprises the elements formed from their coefficients.
For example – for \[n=2\]
We have the matrix of the order \[3\times 3\], the matrix looks like,
\[\left( \begin{matrix}
   {{a}_{2}} & {{a}_{1}} & {{a}_{0}} \\
   2{{a}_{2}} & {{a}_{1}} & 0 \\
   0 & 2{{a}_{2}} & {{a}_{1}} \\
\end{matrix} \right)\]
Then, the discriminant \[\vartriangle \] is given in terms of this Sylvester matrix by the formula,
\[\vartriangle =\dfrac{{{\left( -1 \right)}^{\dfrac{1}{2}n(n-1)}}}{{{a}_{n}}}\left| {{S}_{n}} \right|\]
Where \[{{S}_{n}}\] is the Sylvester matrix.
So, we have,
\[\vartriangle =\dfrac{\left( -1 \right)}{{{a}_{2}}}\left( \begin{matrix}
   {{a}_{2}} & {{a}_{1}} & {{a}_{0}} \\
   2{{a}_{2}} & {{a}_{1}} & 0 \\
   0 & 2{{a}_{2}} & {{a}_{1}} \\
\end{matrix} \right)=a_{1}^{2}-4{{a}_{2}}{{a}_{0}}\]
It is similar to what we know as, \[\vartriangle ={{b}^{2}}-4ac\]
This is similar for other values of \[n\] as well.
Values for \[n=2\] and \[n=3\] should be known as these are the commonly used polynomials. And the respective value can tell about the zeroes of the polynomial, that is, its characteristics whether it is real or not (imaginary).

Note: The formula for the determinant of a polynomial should be correctly written. Students usually get confused with the signs in the numerator and the denominator, so should be taken care of. The value should be calculated step wise to be free of errors.