# How do you find the discriminant of x^7-3x^5+x^4-4x^2+4x+4 and what does it tell us?

Sep 24, 2016

$\Delta = 0$ meaning that there is at least one repeated root, which may be Real or non-Real Complex.

#### Explanation:

The discriminant of a septic polynomial of the form:

${a}_{7} {x}^{7} + {a}_{6} {x}^{6} + {a}_{5} {x}^{5} + {a}_{4} {x}^{4} + {a}_{3} {x}^{3} + {a}_{2} {x}^{2} + {a}_{1} x + {a}_{0}$

is given by the formula

$\Delta = - \frac{1}{a} _ 7 \left\mid \begin{matrix}{a}_{7} & {a}_{6} & {a}_{5} & {a}_{4} & {a}_{3} & {a}_{2} & {a}_{1} & {a}_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & {a}_{7} & {a}_{6} & {a}_{5} & {a}_{4} & {a}_{3} & {a}_{2} & {a}_{1} & {a}_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & {a}_{7} & {a}_{6} & {a}_{5} & {a}_{4} & {a}_{3} & {a}_{2} & {a}_{1} & {a}_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & {a}_{7} & {a}_{6} & {a}_{5} & {a}_{4} & {a}_{3} & {a}_{2} & {a}_{1} & {a}_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & {a}_{7} & {a}_{6} & {a}_{5} & {a}_{4} & {a}_{3} & {a}_{2} & {a}_{1} & {a}_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & {a}_{7} & {a}_{6} & {a}_{5} & {a}_{4} & {a}_{3} & {a}_{2} & {a}_{1} & {a}_{0} \\ 7 {a}_{7} & 6 {a}_{6} & 5 {a}_{5} & 4 {a}_{4} & 3 {a}_{3} & 2 {a}_{2} & {a}_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 7 {a}_{7} & 6 {a}_{6} & 5 {a}_{5} & 4 {a}_{4} & 3 {a}_{3} & 2 {a}_{2} & {a}_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 7 {a}_{7} & 6 {a}_{6} & 5 {a}_{5} & 4 {a}_{4} & 3 {a}_{3} & 2 {a}_{2} & {a}_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 7 {a}_{7} & 6 {a}_{6} & 5 {a}_{5} & 4 {a}_{4} & 3 {a}_{3} & 2 {a}_{2} & {a}_{1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 7 {a}_{7} & 6 {a}_{6} & 5 {a}_{5} & 4 {a}_{4} & 3 {a}_{3} & 2 {a}_{2} & {a}_{1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 7 {a}_{7} & 6 {a}_{6} & 5 {a}_{5} & 4 {a}_{4} & 3 {a}_{3} & 2 {a}_{2} & {a}_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 7 {a}_{7} & 6 {a}_{6} & 5 {a}_{5} & 4 {a}_{4} & 3 {a}_{3} & 2 {a}_{2} & {a}_{1}\end{matrix} \right\mid$

In our example, $f \left(x\right) = {x}^{7} - 3 {x}^{5} + {x}^{4} - 4 {x}^{2} + 4 x + 4$

we have:

$\left\{\begin{matrix}{a}_{7} = 1 \\ {a}_{6} = 0 \\ {a}_{5} = - 3 \\ {a}_{4} = 1 \\ {a}_{3} = 0 \\ {a}_{2} = - 4 \\ {a}_{1} = 4 \\ {a}_{0} = 4\end{matrix}\right.$

So using some row and column operations we find:
$\Delta = - \frac{1}{1} \left\mid \begin{matrix}1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 \\ 7 & 0 & - 15 & 4 & 0 & - 8 & 4 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 & 0 & 0 \\ 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 & 0 \\ 0 & 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 \\ 0 & 0 & 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 \\ 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 & 0 & 0 & 0 & 0 \\ 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 & 0 & 0 & 0 \\ 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 \\ 0 & 0 & 0 & 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 & 0 \\ 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 \\ 0 & 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 \\ - 3 & 18 & 14 & - 24 & - 4 & - 24 & - 24 & 0 & 0 & 0 \\ 0 & - 3 & 18 & 14 & - 24 & - 4 & - 24 & - 24 & 0 & 0 \\ 0 & 0 & - 3 & 18 & 14 & - 24 & - 4 & - 24 & - 24 & 0 \\ 0 & 0 & 0 & - 3 & 18 & 14 & - 24 & - 4 & - 24 & - 24 \\ 0 & 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 \\ 0 & 0 & 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 \\ 0 & 0 & 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 & 0 \\ 0 & 1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 \\ 18 & 5 & - 21 & - 4 & - 36 & - 12 & 12 & 0 & 0 \\ 0 & 18 & 5 & - 21 & - 4 & - 36 & - 12 & 12 & 0 \\ 0 & 0 & 18 & 5 & - 21 & - 4 & - 36 & - 12 & 12 \\ 0 & 0 & - 3 & 18 & 14 & - 24 & - 4 & - 24 & - 24 \\ 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 \\ 0 & 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 \\ 0 & 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}1 & 0 & - 3 & 1 & 0 & - 4 & 4 & 4 \\ 5 & 33 & - 22 & - 36 & 60 & - 60 & - 72 & 0 \\ 0 & 5 & 33 & - 22 & - 36 & 60 & - 60 & - 72 \\ 0 & 18 & 5 & - 21 & - 4 & - 36 & - 12 & 12 \\ 0 & - 3 & 18 & 14 & - 24 & - 4 & - 24 & - 24 \\ 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 \\ 0 & 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 \\ 0 & 7 & 0 & - 15 & 4 & 0 & - 8 & 4\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}33 & - 7 & - 41 & 60 & - 40 & - 92 & - 20 \\ 5 & 33 & - 22 & - 36 & 60 & - 60 & - 72 \\ 18 & 5 & - 21 & - 4 & - 36 & - 12 & 12 \\ - 3 & 18 & 14 & - 24 & - 4 & - 24 & - 24 \\ 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 \\ 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 \\ 7 & 0 & - 15 & 4 & 0 & - 8 & 4\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}1 & - 12 & 10 & 56 & - 4 & - 64 & - 40 \\ 5 & 33 & - 22 & - 36 & 60 & - 60 & - 72 \\ 18 & 5 & - 21 & - 4 & - 36 & - 12 & 12 \\ - 3 & 18 & 14 & - 24 & - 4 & - 24 & - 24 \\ 6 & - 3 & 0 & 20 & - 24 & - 28 & 0 \\ 0 & 6 & - 3 & 0 & 20 & - 24 & - 28 \\ 7 & 0 & - 15 & 4 & 0 & - 8 & 4\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}93 & - 72 & - 316 & 80 & 260 & 128 \\ 221 & - 201 & - 1012 & 36 & 1140 & 732 \\ - 18 & 44 & 144 & - 16 & - 216 & - 144 \\ 69 & - 60 & - 316 & 0 & 356 & 240 \\ 6 & - 3 & 0 & 20 & - 24 & - 28 \\ 84 & - 85 & - 388 & 28 & 440 & 276\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}1 & 31 & 136 & 36 & - 276 & - 204 \\ 221 & - 201 & - 1012 & 36 & 1140 & 732 \\ - 18 & 44 & 144 & - 16 & - 216 & - 144 \\ 69 & - 60 & - 316 & 0 & 356 & 240 \\ 6 & - 3 & 0 & 20 & - 24 & - 28 \\ 84 & - 85 & - 388 & 28 & 440 & 276\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}- 7052 & - 31068 & - 7920 & 62136 & 45816 \\ 602 & 2592 & 632 & - 5184 & - 3816 \\ - 2199 & - 9700 & - 2484 & 19400 & 14316 \\ - 189 & - 816 & - 196 & 1632 & 1196 \\ - 2689 & - 11812 & - 2996 & 23624 & 17412\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = - \left\mid \begin{matrix}- 7052 & - 31068 & - 7920 & 0 & 45816 \\ 602 & 2592 & 632 & 0 & - 3816 \\ - 2199 & - 9700 & - 2484 & 0 & 14316 \\ - 189 & - 816 & - 196 & 0 & 1196 \\ - 2689 & - 11812 & - 2996 & 0 & 17412\end{matrix} \right\mid$

$\textcolor{w h i t e}{\Delta} = 0$

If $\Delta > 0$ then our septic polynomial would have $0$ or $1$ Complex conjugate pairs of zeros and $7$ or $5$ Real zeros.

If $\Delta < 0$ then our septic polynomial would have $2$ or $3$ Complex conjugate pairs of zeros and $3$ or $1$ Real zeros.

Since $\Delta = 0$, our septic polynomial has at least one repeated root which may or may not be Real.