CLISP Polynomial Manipulation

[Seldar]

Hello everyone and Merry Christmas.

The reason I am opening this thread is that I am required to write a polynomial calculator in Common LISP as part of a Course Work. As I am newbie to LISP(s) I am asking you if anyone can help me with it . The calculator should have 3 basic functions(p+ p- p*) each of which should be of exactly 2 arguments, such as (p+ arg1 arg2) where arg1 can be for example 1+x and arg2 can be 2*x then the result should be 1+x+2*x = 1+3*x. To sum up
the program should be able to behave: (p+ (p+ 1 x) (p* 2 x)) >> 1 + 3x

; Constructors

(defun make-constant (num)
  num)

(defun make-variable (sym)
  sym)
(defmacro p+ (poly1 poly2)
`(list '+ ',poly1 ',poly2))

(defmacro p* (poly1 poly2)
`(list '* ',poly1 ',poly2))

defun make-power (poly num)
  (list '** poly num))

defun p- (poly num)
  (list '- ',poly1 ',poly2))

;; Recognizers for polynomials

(defun constant-p (poly)
  (numberp poly))

(defun variable-p (poly)
  (symbolp poly))

(defun sum-p (poly)
  (and (listp poly) (eq (first poly) '+)))

(defun subtr-p (poly)
  (and (listp poly) (eq (first poly) '-)))

(defun product-p (poly)
  (and (listp poly) (eq (first poly) '*)))

(defun power-p (poly)
  (and (listp poly) (eq (first poly) '**)))

;; Selectors for polynomials

(defun constant-numeric (const)
  const)

(defun variable-symbol (var)
  var)

(defun sum-arg1 (sum)
  (second sum))

(defun sum-arg2 (sum)
  (third sum))

(defun product-arg1 (prod)
  (second prod))

(defun product-arg2 (prod)
  (third prod))

(defun power-base (pow)
  (second pow))

(defun power-exponent (pow)
  (third pow))

;; Simplification function

(defun simplify (poly)
  "Simplify polynomial POLY."
  (cond
   ((constant-p poly) poly)
   ((variable-p poly) poly)
   ((sum-p poly)
    (let ((arg1 (simplify (sum-arg1 poly)))
	  (arg2 (simplify (sum-arg2 poly))))
      (make-simplified-sum arg1 arg2)))
   ((product-p poly)
    (let ((arg1 (simplify (product-arg1 poly)))
	  (arg2 (simplify (product-arg2 poly))))
      (make-simplified-product arg1 arg2)))
   ((power-p poly)
    (let ((base (simplify (power-base poly)))
	  (exponent (simplify (power-exponent poly))))
      (make-simplified-power base exponent)))
   ((derivative-p poly) poly)))

(defun make-simplified-sum (arg1 arg2)
  "Given simplified polynomials ARG1 and ARG2, construct a simplified sum of ARG1 and ARG2."
  (cond
   ((and (constant-p arg1) (zerop arg1)) arg2)
   ((and (constant-p arg2) (zerop arg2)) arg1)
   (t                                    (p+ arg1 arg2))))

(defun make-simplified-product (arg1 arg2)
  "Given simplified polynomials ARG1 and ARG2, construct a simplified product of ARG1 and ARG2."
  (cond
   ((and (constant-p arg1) (zerop arg1)) (make-constant 0))
   ((and (constant-p arg2) (zerop arg2)) (make-constant 0))
   ((and (constant-p arg1) (= arg1 1))   arg2)
   ((and (constant-p arg2) (= arg2 1))   arg1)
   (t                                    (p* arg1 arg2))))

(defun make-simplified-power (base exponent)
  "Given simplified polynomials BASE and EXPONENT, construct a simplified power with base BASE and exponent EXPONENT."
  (cond
   ((and (constant-p exponent) (= exponent 1))   base)
   ((and (constant-p exponent) (zerop exponent)) (make-constant 1))
   (t                          (make-power base exponent))))

This is the code, but it is not working properly. I need your help to make it work. Thank you in advance.