Convert the following Scheme/Racket synthetic division function to not use assignment. (Meaning no Set!). Must use the base Racket/Scheme libraries. #lang racket/base (require racket/list) ;; dividend and divisor are both polynomials, which are here simply lists of coefficients. ;; Eg: x^2 + 3x + 5 will be represented as (list 1 3 5) (define (extended-synthetic-division dividend divisor) (define out (list->vector dividend)) ; Copy the dividend ;; for general polynomial division (when polynomials are non-monic), we need to normalize by ;; dividing the coefficient with the divisor's first coefficient (define normaliser (car divisor)) (define divisor-length (length divisor)) ; } we use these often enough (define out-length (vector-length out)) ; } (for ((i (in-range 0 (- out-length divisor-length -1)))) (vector-set! out i (quotient (vector-ref out i) normaliser)) (define coef (vector-ref out i)) (unless (zero? coef) ; useless to multiply if coef is 0 (for ((i+j (in-range (+ i 1) ; in synthetic division, we always skip the first (+ i divisor-length))) ; coefficient of the divisior, because it's (divisor_j (in-list (cdr divisor)))) ; only used to normalize the dividend coefficients (vector-set! out i+j (+ (vector-ref out i+j) (* coef divisor_j -1)))))) ;; The resulting out contains both the quotient and the remainder, the remainder being the size of ;; the divisor (the remainder has necessarily the same degree as the divisor since it's what we ;; couldn't divide from the dividend), so we compute the index where this separation is, and return ;; the quotient and remainder. ;; return quotient, remainder (conveniently like quotient/remainder) (split-at (vector->list out) (- out-length (sub1 divisor-length)))) 

;Code for testing/OUTPUT (module+ main (displayln "POLYNOMIAL SYNTHETIC DIVISION") (define N '(1 -12 0 -42)) (define D '(1 -3)) (define-values (Q R) (extended-synthetic-division N D)) (printf "~a / ~a = ~a remainder ~a~%" N D Q R))

Console Output: POLYNOMIAL SYNTHETIC DIVISION (1 -12 0 -42) / (1 -3) = (1 -9 -27) remainder (-123)