module Domain.Math.Polynomial.Examples
( linearExamples, quadraticExamples, higherDegreeExamples
, factorizeExamples, expandExamples
, ineqLin1, ineqQuad1, ineqQuad2, extraIneqQuad, ineqHigh
) where
import Data.Monoid
import Domain.Math.Data.Relation
import Domain.Math.Expr
import Ideas.Common.Exercise
import Ideas.Common.Rewriting
import Prelude hiding ((^))
x :: Expr
x = variable "x"
linearExamples :: Examples (Equation Expr)
linearExamples =
examplesFor Easy
[ 5*x + 3 :==: 18
, 11*x - 12 :==: 21
, 19 - 3*x :==: -5
, -12 + 5*x :==: 33
, 15 - 9*x :==: 6
, 4*x + 18 :==: 0
, 11*x - 12 :==: -34
, -2*x - 3 :==: -4
, 6*x - 12 :==: 2
, -4*x - 13 :==: -11
] <>
examplesFor Easy
[ 6*x-2 :==: 2*x+14
, 3+6*x :==: 3*x+24
, 5*x+7 :==: 2*x - 10
, 2*x-8 :==: 18 - x
, 4*x - 6 :==: 7*x - 14
, -1 -5*x :==: 3*x - 20
, 4*x - 7 :==: -5*x - 24
, 4*x - 18 :==: 14 + 11*x
, 17 :==: 4 - 10*x
, -5*x + 6 :==: 2 - 3*x
] <>
examplesFor Medium
[ 4*(x-1) :==: 11*x - 12
, 4*(x-4) :==: 5*(2*x+1)
, 2*(5-3*x) :==: 6-x
, 4*x - (x-2) :==: 12 + 5*(x-1)
, -3*(x-2) :==: 3*(x+4) - 7
, 3*(4*x-1) + 3 :==: 7*x - 14
, 4*(4*x - 1) - 2 :==: -3*x + 3*(2*x -5)
, 2*x - (3*x + 5) :==: 10 + 5*(x-1)
, -5*(x+1) :==: 9*(x+4)-5
, 18 - 2*(4*x + 2) :==: 7*x - 4*(4*x -2)
] <>
examplesFor Medium
[ (1/2)*x - 4 :==: 2*x + mixed 2 1 2
, (1/4)*x + (1/2) :==: (5/2)*x + 2
, (1/4)*x - (3/4) :==: 2*x + (1/2)
, -(1/2)*x + (3/4) :==: (5/2)*x + 3
, -(1/2)*x + mixed 1 1 2 :==: 2*x - 5
, -(1/3)*x + (3/4) :==: (1/4)*x + (1/6)
, (3/4)*x - (1/3) :==: (2/3)*x - (3/4)
, (2/5)*x - (1/4) :==: (1/2)*x + (3/4)
, (2/3)*x - 2 :==: (1/5)*x - (3/5)
, (-mixed 1 2 5)*x + mixed 3 1 2 :==: (3/5)*x + (9/10)
] <>
examplesFor Medium
[ (1/4)*(x-3) :==: (1/2)*x - 4
, (x+3)/2 :==: 5*((1/2)*x + mixed 1 1 2)
, (1/2)*(7-(2/3)*x) :==: 2 + (1/9)*x
, (3/4)*x - (x-1) :==: 3 + mixed 2 1 2*(x-1)
, -(5/4)*(x-7) :==: (3/4)*(x+2) - mixed 4 1 2
, 3*((1/5)*x - 1) + 5 :==: 7*x - 14
, ((5*x - 1) / 6) - 2 :==: -4*x + (3*x - 6)/2
, 2*x - ((2*x+2)/5) :==: 12 + (x-1)/6
, (-3*(x+2))/6 :==: 9*((2/3)*x + (1/3)) - (5/3)
, 1 - ((4*x + 2)/3) :==: 3*x - ((5*x - 1) / 4)
]
quadraticExamples :: Examples (Equation Expr)
quadraticExamples =
examplesFor Easy
[ x^2 :==: 2
, x^2+3 :==: 52
, x^2-7 :==: 0
, 9*x^2 - 6 :==: 75
, 32 - 2*x^2 :==: 14
, 2*(x^2 - 3) :==: 12
, (1/4)*x^2 + 12 :==: 16
, (x-1)^2 :==: 100
, 14 - 2*x^2 :==: 6
, (1/4)*(17-x^2) :==: 2
] <>
examplesFor Medium
[ (x-7)^2 + 3 :==: 11
, (6-2*x)^2 :==: 81
, (1/2)*(x+9)^2 :==: 4
, (3-x^2)/10 :==: 2
, 5*x^2 + 3*x :==: 3*x + 2
, 11 - (2*x + 1)^2 :==: 5
, (6*x - 3)^2 + 6 :==: 12
, (7+2*x)^2 :==: 7
, 4 - (x^2 / 10) :==: 6
, 12 - (2*x + 3)^2 :==: 6
] <>
examplesFor Medium
[ x^2 :==: 5*x
, x^2 - 6*x :==: 0
, 6*x + x^2 :==: 0
, x*(x+4) :==: 0
, x*(2*x-4) :==: 0
, 3*x^2 :==: 6*x
, 3*x :==: 2*x^2
, x*(1-6*x) :==: 0
, (x+5)*(x-8) :==: 0
, (3*x-1)*(x+3) :==: 0
] <>
examplesFor Medium
[ x^2-2*x :==: 3
, x^2+12*x+20 :==: 0
, x^2-x :==: 30
, x*(x+2) :==: 8
, x*(x-3) :==: 4
, 2*x+15 :==: x^2
, 4*x :==: 12 - x^2
, x^2 :==: 15 - 8*x
, x^2-9*x+18 :==: 0
, x^2+14*x+24 :==: 0
] <>
examplesFor Difficult
[ (3*x+5)^2+(x-5)^2 :==: 40
, 4*(10-x^2) :==: -2*x*(2*x + 10)
, x*(x+12) :==: 2*x^2
, 3*(x-2)*(x+6) :==: 12*x
, 8*x^2+4*x-24 :==: (x+3)*(x-8)
, 3*x^2 - 11 :==: (3+2*x)^2
, 2*x*(x-3)-3 :==: (x+2)*(x+6)
, 12*(x^2-3*x)+8 :==: 56
, 4*x^2-6*x :==: x^2+9
, (x+1)*(x-5) :==: (x-2)*(x-3)
] <>
examplesFor Difficult
[ x^2+4*x-4 :==: 0
, x^2-6*x :==: 4
, x^2-12*x+34 :==: 0
, 2*x^2+4*x-8 :==: 0
, (x-4)*(x-1) :==: 11
, (x-(7/2))^2 :==: 2*(x+4)
, x^2-3*x :==: 3*(x-2)
, (4-x)*(1-x) :==: 3*x
, 2*x^2 :==: x*(x+2)+7
, (1-x)^2 :==: x+2
]
factorizeExamples :: Examples Expr
factorizeExamples =
examplesFor Easy
[
4*x^2 -4*x
, 36*x^2+30*x
, -6*x^2-18*x
, 14*x^2-10*x
, x^2+11*x+24
, x^2-8*x+15
, x^2-x-2
, x^2-11*x+28
]
expandExamples :: Examples Expr
expandExamples = examplesFor Easy
[ 5*(x+1), -3*(x-3), (x-1)*7
, 4*(3-2*x), (x+1)*(x-3), (x+1)*(1-x)
, x*(x-1), 3*(x-2)*2*x
, (x-1)^2, (x+1)^2, (x-1)^2*(x+1)
, (x+1)^3, (x-1)^3*x, (x-1)*(x+3)*(x-5)
, x/2, (x+1)/2, (x+1)/2 + (x+2)/3, ((x+1)/2) * ((x+2)/3)
]
higherDegreeExamples :: Examples (Equation Expr)
higherDegreeExamples =
examplesFor Easy
[
(1/3)*x^3 :==: 9
, x^5 - 12 :==: 20
, 1 - 8*x^3 :==: -124
, 16 - 32*x^5 :==: - 227
, 3*x^4 :==: 48
, (1/9)*x^6 + 12 :==: 93
, 39 - 8*x^2 :==: 21
, (1/2)*x^4 - 13 :==: 27.5
, 3*(2*x-1)^3 + 11 :==: 659
, 0.5*(3*x-4)^5 + 7 :==: 23
, 2*(0.5*x+3)^7 - 11 :==: -9
, 5*(1-4*x)^3 + 4 :==: -621
, 3*(2*x + 5)^2 + 9 :==: 21
, 2*(3*x-6)^6 - 24 :==: -22
, -2*(4*x-5)^4 + 192 :==: -8000
, (3-2*x)^4 + 23 :==: 279
] <>
examplesFor Easy
[
2*x^3 + 9 :==: 19
, 4*x^5 - 17 :==: 27
, 3*x^7 + 9 :==: 62
, 5*x^3 - 1 :==: 9
, 6 - 5*x^3 :==: 76
, 11 - 7*x^5 :==: 53
, 4 - 0.2*x^7 :==: 9
, 18 - 11*x^7 :==: 62
, 0.5*x^4 + 5 :==: 12
, 5*x^6 - 37 :==: 68
, 4*x^8 - 19 :==: 9
, 5*x^6 + 7 :==: 97
, 18 - 7*x^4 :==: -38
, 3 + (1/3)*x^6 :==: 7
, 1 - (1/9)*x^8 :==: -4
, 47 + 15*x^8 :==: 77
, 18*x^8 - 11 :==: 7
, (1/4)*x^6 + 14 :==: 30
, 5*x^4 + 67 :==: 472
, 5*x^4 - 1 :==: 4
, (1/8)*x^7 + 24 :==: 40
, 0.2*x^3 + 27 :==: 52
, 32*x^3 + 18 :==: 22
, 4*x^3 - 8 :==: 100
, 14 -2*x^3 :==: 700
, 4-3*x^5 :==: 100
, 14 - 11*x^7 :==: 25
, 1 - 3*x^5 :==: 97
, 3*(x-2)^4 + 7 :==: 37
, 6 - (2*x-1)^3 :==: 1
, (1/3)*(x+5)^6 - 4 :==: 3
, 6 - 0.5*(x-1)^5 :==: 10
, (1/2)*(3*x-1)^4 :==: 8
, 100-(1/3)*(4*x-3)^5 :==: 19
, 4*(0.5*x+2)^6 + 5 :==: 9
, 3*(2*x + 7)^3 + 11 :==: 35
] <>
examplesFor Medium
[
x^3 - 5*x^2 + 4*x :==: 0
, x^3 :==: 3*x^2 + 10*x
, 14*x :==: x^3 + 5*x^2
, (1/2)*x^3 + 3*x^2 + 4*x :==: 0
, x^3 + 6*x^2 + 9*x :==: 0
, 5*x^2 :==: x^3 + 6*x
, x^3 - 5*x^2 :==: 6*x
, x^3 :==: 4*x^2 + 12*x
, x^4 + 36 :==: 13*x^2
, x^4 - 9*x^2 + 20 :==: 0
, x^4 :==: 2*x^2 + 3
, x^4 + 2*x^2 :==: 24
, 7*x^2 + 18 :==: x^4
, x^4 :==: x^2 + 12
, 29*x^2 :==: x^4 + 100
, 2*x^4 + 2*x^2 :==: 12
, 4*x^4 + 4 :==: 17*x^2
, 16*x^4 + 225 :==: 136*x^2
, 2*x^4 - 15*x^2 + 25 :==: 0
, 9*x^4 - 28*x^2 + 3 :==: 0
, 3*x^4 - 14*x^2 - 5 :==: 0
, 2*x^4 :==: x^2 + 3
, 9*x^4 + 14*x^2 :==: 8
, 4*x^4 - 29*x^2 - 24 :==: 0
, 8*x^6 - 9*x^3 + 1 :==: 0
, 27*x^6 + 8 :==: 217*x^3
, 2*x^6 + x^3 - 1 :==: 0
, 8*x^6 + 31*x^3 :==: 4
, 3*x^6 - 80*x^3 - 27 :==: 0
, 5*x^6 :==: 39*x^3 + 8
, 7*x^6 + 8*x^3 + 1 :==: 0
, 4*x^6 + 2 :==: -9*x^3
] <>
examplesFor Difficult
[ x^3 + x^2 :==: 0
, x^3 - 5*x :==: 0
, x^3 - 11*x^2 + 18*x :==: 0
, x^3 + 36*x :==: 13*x^2
, x^3 + 2*x^2 :==: 24*x
, 7*x^3 :==: 8*x^2
, x^4 :==: 9*x^2
, 64*x^7 :==: x^5
, x^3 - 4*x^2 - 9*x :==: 0
, (x-1)*(x^3 - 6*x) :==: 3*x^3 - 3*x^2
]
ineqLin1 :: [[Inequality Expr]]
ineqLin1 =
let a = Var "a" in
[ [ 7*x - 12 :<: 5*x + 3
, 4*(x-3) :>: 3*(x-4)
, 6*(a+1) :<: 3*(a-2)+4
, 5 - 2*(a-3) :>: 5*(3-a)
]
, [ 4*x+5 :<: 5*x - 3
, (1/3)*x+10 :>: (1/2)*x
, 3*x+1 :<: 7*x + 5
, x+6 :>: 2 - (3/4)*x
]
, [ 5*(x-1) :<: 7*x - 1
, -3*(4*x-1) :>: 2-(x-1)
, 2*(3*x-1) :<: 5-(2-9*x)
, 2*(x-1)-3*(x-2) :>: 6
]
]
ineqQuad1 :: [[Inequality Expr]]
ineqQuad1 =
[ [ x^2 +3*x-4 :<: 0
, x^2-4*x-12 :>: 0
, -x^2 - 4*x + 5 :<: 0
, -x^2 + 3*x + 18 :>: 0
, (1/2)*x^2 - 3*x - 8 :<: 0
, -2*x^2 + 10*x :>: 0
]
, [ x^2 + 9*x :<: 3*x - 5
, x^2 - x :>: 12
, x^2 - 4.5*x :<: 7-3*x
, 2*x^2 - 10*x :>: x^2 - 3*x
, 4*x^2 + 6*x :<: x^2 + 3*x + 18
, 2*x^2 + 6*x - 10 :>: x^2 + 2*x - 5
]
]
ineqHigh :: [Inequality Expr]
ineqHigh =
[ 2*x^3 :>: 54
, -0.5*x^4 :<: -40.5
, 1 - 2*x^5 :<: -485
, (2*x-3)^4 :>: 1
, -(0.5*x+2)^3 :<: -1
, 0.25*(0.5*x-2)^4 :<: 4
]
ineqQuad2 :: [Inequality Expr]
ineqQuad2 =
[ x^2 + 9*x :<: 3*x - 5
, x^2 - x :>: 12
, x^2 - 4.5*x :<: 7 - 3*x
, 2*x^2 - 10*x :>: x^2 - 3*x
, 4*x^2 + 6*x :<: x^2 + 3*x + 18
, 2*x^2 + 6*x - 10 :>: x^2 +2*x - 5
]
extraIneqQuad :: [Inequality Expr]
extraIneqQuad =
[ x^2-x-7 :>: -100, x^2-x-7 :<: -100, x^2 :<: x^2, x :>=: x
, x^2 :>=: 0, x^2 :>: 0, x^2 :<: 0, x^2 :<=: 0
]