Question

factorise 1331t² - 7260t²s + 13200ts² + 8000s³​

Answer 1

Input:

1331 t^2 - 7260 t^2 s + 13200 t s^2 + 8000 s^3

3D plot:

3D plot

Contour plot:

Contour plot

Alternate forms:

8000 s^3 + 13200 s^2 t + 121 (11 - 60 s) t^2

8000 s^3 + t (13200 s^2 + (1331 - 7260 s) t)

s (s (8000 s + 13200 t) - 7260 t^2) + 1331 t^2

Roots:

60 s - 11!=0, t = (40 (15 s^2 - sqrt(5) sqrt(s^3 (105 s - 11))))/(11 (60 s - 11))

60 s - 11!=0, t = (40 (sqrt(5) sqrt(s^3 (105 s - 11)) + 15 s^2))/(11 (60 s - 11))

Root:

s = 11/60, t = -1/9

Polynomial discriminant:

Δ_s = 3061257408000000 (7 t^6 - 10 t^5 - t^4)

Integer root:

s = 0, t = 0

Roots for the variable t:

t = (40 (15 s^2 - sqrt(5) sqrt(105 s^4 - 11 s^3)))/(11 (60 s - 11))

t = (40 (15 s^2 + sqrt(5) sqrt(105 s^4 - 11 s^3)))/(11 (60 s - 11))

Derivative:

d/ds(1331 t^2 - 7260 t^2 s + 13200 t s^2 + 8000 s^3) = 60 (400 s^2 + 440 s t - 121 t^2)

Indefinite integral:

integral(8000 s^3 + 13200 s^2 t + 1331 t^2 - 7260 s t^2) ds = 2000 s^4 + 4400 s^3 t - 3630 s^2 t^2 + 1331 s t^2 + constant

Definite integral over a disk of radius R:

integral integral_(s^2 + t^2<R^2)(8000 s^3 + 13200 s^2 t - 7260 s t^2 + 1331 t^2) ds dt = (1331 π R^4)/4

Definite integral over a square of edge length 2 L:

integral_(-L)^L integral_(-L)^L (8000 s^3 + 13200 s^2 t + 1331 t^2 - 7260 s t^2) dt ds = (5324 L^4)/3