Lisp Polynomial manipulation

[nakias]

Hello,

I need your help for the following application
The app has to do with polynomial manipulation and my question is how can i manipulate both letters and numbers.
in other words how can i collect together terms , e.g., (x + 1) + x → 2x + 1 rather than x + 1 + x

Thanks in advance for any help !

;; Constructors

(defun make-constant (num)
  num)

(defun make-variable (sym)
  sym)

;;(defun p+ (poly1 poly2)
 ;; (list '+ poly1 poly2))

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

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


;;(defun p* (poly1 poly2)
;;  (list '* poly1 poly2))

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

;;
;; 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 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))))

[Rashakil Fol]

Sort them by the kind of terms they have, and then group the similar terms together.

Alternately, don't use letters at all in your representation of polynomials -- but this only works if you limit yourself to polynomials of a single variable.

[nakias]

Is there anything modifiable on the above code that can do the work ? :S
Actually the code accepts the polynomial but it doesn't do any computation:
given (p+ 4 (p+ 4 x))
return (+ 4 (P+ 4 X))