-- | First assignment for IB016, semester spring 2015
-- Name: Name Surname
-- UID: 123456

module Polynomial (
    -- * Polynomial type
      Polynomial (..)
    -- * Printing
    , polyPrint
    -- * Evaluation
    , eval
    , belongs
    -- * Properties
    , yIntersect
    , quadratic
    -- * Operations
    , scalarTimes
    , polyPlus
    , polyMinus
    , polyTimes
    ) where

-- | The Polynomial type. Polynomials are represented as a list of coefficients
-- of increasing power. That is, the first element of list is constant element,
-- the second is the coefficient for x, the third coefficient for x^2 and so on.
-- There should be no trailing zeroes in the polynomial representation, i.e.
-- x^2 + 2x + 3 is represented as @'P' [3,2,1]@ not @'P' [3,2,1,0,0]@ or similar.
newtype Polynomial a = P { unP :: [a] } deriving Show

-- | Evaluate polynomial in given value. This can be viewed as
-- converting the polynomial to the evaluation function.
--
-- >>> eval (P [3, 1, -5]) 2
-- -15
eval :: Num a => Polynomial a -> (a -> a)
eval = undefined

-- | Determines, if a given point belongs to the polynomial.
--
-- >>> belongs (2, -15) (P [3, 1, -5])
-- True
belongs :: (Num a, Eq a) => (a, a) -> Polynomial a -> Bool
belongs = undefined

-- | Determines, if the polynomial is quadratic
--
-- >>> quadratic (P [1, 2, -1])
-- True
quadratic :: Polynomial a -> Bool
quadratic = undefined

-- | Find the coordinates of the polynomial intersecting the y axis.
--
-- >>> yIntersect (P [1, 2, -1])
-- (0,1)
yIntersect :: Num a => Polynomial a -> (a, a)
yIntersect = undefined

-- | Print polynomial into pretty string representation.
--
-- The first argument is the variable name. Write spaces in between the term
-- operators. Coefficients 1 and -1 should be diplayed only as a sign symbol.
-- Do not display terms with zero coefficients.
--
-- >>> polyPrint "x" (P [1, 2, 3, 4])
-- "4x^3 + 3x^2 + 2x + 1"
-- >>> polyPrint "x" (P [-1, 0, -1, 3, -4])
-- "- 4x^4 + 3x^3 - x^2 - 1"
-- >>> polyPrint "x" (P [0, 0, -1])
-- "- x^2"
-- >>> polyPrint "x" (P [])
-- "0"
polyPrint :: (Num a, Ord a, Show a) => String -> Polynomial a -> String
polyPrint = undefined

-- | Multiply scalar and polynomial.
--
-- >>> scalarTimes 3 (P [1, 2, 3])
-- P [3, 6, 9]
-- >>> scalarTimes 0 (P [1, 2, 3])
-- P []
scalarTimes :: (Eq a, Num a) => a -> Polynomial a -> Polynomial a
scalarTimes = undefined

-- | Sum two polynomials.
--
-- >>> polyPlus (P [1, 2, 3]) (P [2, 2, -3])
-- P [3, 4]
polyPlus :: (Num a, Eq a) => Polynomial a -> Polynomial a -> Polynomial a
polyPlus = undefined

-- | Make a difference of two polynomials.
--
-- >>> polyMinus (P [2, -5, 3, 3]) (P [1, 2, 3])
-- P [1, -7, 0, 3]
polyMinus :: (Num a, Eq a) => Polynomial a -> Polynomial a -> Polynomial a
polyMinus = undefined

-- | Multiply two polynomials.
--
-- >>> polyTimes (P [2, 1]) (P [2, 3])
-- P [4, 8, 3]
polyTimes :: (Num a, Eq a) => Polynomial a -> Polynomial a -> Polynomial a
polyTimes = undefined