Question

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

Answer 1

`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 = -1/a_7 abs((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),(7a_7, 6a_6, 5a_5, 4a_4, 3a_3, 2a_2, a_1, 0, 0, 0, 0, 0, 0),(0, 7a_7, 6a_6, 5a_5, 4a_4, 3a_3, 2a_2, a_1, 0, 0, 0, 0, 0),(0, 0, 7a_7, 6a_6, 5a_5, 4a_4, 3a_3, 2a_2, a_1, 0, 0, 0, 0),(0, 0, 0, 7a_7, 6a_6, 5a_5, 4a_4, 3a_3, 2a_2, a_1, 0, 0, 0),(0, 0, 0, 0, 7a_7, 6a_6, 5a_5, 4a_4, 3a_3, 2a_2, a_1, 0, 0),(0, 0, 0, 0, 0, 7a_7, 6a_6, 5a_5, 4a_4, 3a_3, 2a_2, a_1, 0),(0, 0, 0, 0, 0, 0, 7a_7, 6a_6, 5a_5, 4a_4, 3a_3, 2a_2, a_1))`

In our example, `f(x) = x^7 - 3x^5 + x^4 - 4x^2 + 4x + 4`

we have:

`{ (a_7=1), (a_6=0), (a_5=-3), (a_4=1), (a_3=0), (a_2=-4), (a_1=4), (a_0=4) :}`

So using some row and column operations we find:
`Delta = -1/1 abs((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))`

`color(white)(Delta) = - abs((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))`

`color(white)(Delta) = - abs((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))`

`color(white)(Delta) = - abs((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))`

`color(white)(Delta) = - abs((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))`

`color(white)(Delta) = - abs((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))`

`color(white)(Delta) = - abs((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))`

`color(white)(Delta) = - abs((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))`

`color(white)(Delta) = - abs((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))`

`color(white)(Delta) = - abs((-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))`

`color(white)(Delta) = - abs((-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))`

`color(white)(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.