diff --git a/extra/math/polynomials/polynomials-docs.factor b/extra/math/polynomials/polynomials-docs.factor
new file mode 100644
index 0000000000..08b7ca7c4d
--- /dev/null
+++ b/extra/math/polynomials/polynomials-docs.factor
@@ -0,0 +1,94 @@
+USING: help.markup help.syntax math sequences ;
+IN: math.polynomials
+
+ARTICLE: "polynomials" "Polynomials"
+"A polynomial is a vector with the highest powers on the right:"
+{ $code "{ 1 1 0 1 } -> 1 + x + x^3" "{ } -> 0" }
+"Numerous words are defined to help with polynomial arithmetic:"
+{ $subsection p= }
+{ $subsection p+ }
+{ $subsection p- }
+{ $subsection p* }
+{ $subsection p-sq }
+{ $subsection powers }
+{ $subsection n*p }
+{ $subsection p/mod }
+{ $subsection pgcd }
+{ $subsection polyval }
+{ $subsection pdiff }
+{ $subsection pextend-conv }
+{ $subsection ptrim }
+{ $subsection 2ptrim } ;
+
+ABOUT: "polynomials"
+
+HELP: powers
+{ $values { "n" integer } { "x" number } { "seq" sequence } }
+{ $description "Output a sequence having " { $snippet "n" } " elements in the format: " { $snippet "{ 1 x x^2 x^3 ... }" } "." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "4 2 powers ." "{ 1 2 4 8 }" } } ;
+
+HELP: p=
+{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "?" "a boolean" } }
+{ $description "Tests if two polynomials are equal." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 0 1 } { 0 1 0 } p= ." "t" } } ;
+
+HELP: ptrim
+{ $values { "p" "a polynomial" } { "p" "a polynomial" } }
+{ $description "Trims excess zeros from a polynomial." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 0 1 0 0 } ptrim ." "{ 0 1 }" } } ;
+
+HELP: 2ptrim
+{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "p" "a polynomial" } { "q" "a polynomial" } }
+{ $description "Trims excess zeros from two polynomials." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 0 1 0 0 } { 1 0 0 } 2ptrim swap . ." "{ 0 1 }\n{ 1 }" } } ;
+
+HELP: p+
+{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "r" "a polynomial" } }
+{ $description "Adds " { $snippet "p" } " and " { $snippet "q" } " component-wise." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 0 1 } { 0 1 } p+ ." "{ 1 1 1 }" } } ;
+
+HELP: p-
+{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "r" "a polynomial" } }
+{ $description "Subtracts " { $snippet "q" } " from " { $snippet "p" } " component-wise." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 1 1 } { 0 1 } p- ." "{ 1 0 1 }" } } ;
+
+HELP: n*p
+{ $values { "n" number } { "p" "a polynomial" } { "n*p" "a polynomial" } }
+{ $description "Multiplies each element of " { $snippet "p" } " by " { $snippet "n" } "." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "4 { 3 0 1 } n*p ." "{ 12 0 4 }" } } ;
+
+HELP: pextend-conv
+{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "p" "a polynomial" } { "q" "a polynomial" } }
+{ $description "Convulution, extending to " { $snippet "p_m + q_n - 1" } "." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 0 1 } { 0 1 } pextend-conv swap . ." "V{ 1 0 1 0 }\nV{ 0 1 0 0 }" } } ;
+
+HELP: p*
+{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "r" "a polynomial" } }
+{ $description "Multiplies two polynomials." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 2 3 0 0 0 } { 1 2 0 0 } p* ." "{ 1 4 7 6 0 0 0 0 0 }" } } ;
+
+HELP: p-sq
+{ $values { "p" "a polynomial" } { "p^2" "a polynomial" } }
+{ $description "Squares a polynomial." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 2 0 } p-sq ." "{ 1 4 4 0 0 }" } } ;
+
+HELP: p/mod
+{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "z" "a polynomial" } { "w" "a polynomial" } }
+{ $description "Computes to quotient " { $snippet "z" } " and remainder " { $snippet "w" } " of dividing " { $snippet "p" } " by " { $snippet "q" } "." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 1 1 1 } { 3 1 } p/mod swap . ." "V{ 7 -2 1 }\nV{ -20 0 0 }" } } ;
+
+HELP: pgcd
+{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "a" "a polynomial" } { "d" "a polynomial" } }
+{ $description "Computes the greatest common divisor " { $snippet "d" } " of " { $snippet "p" } " and " { $snippet "q" } ", and another value " { $snippet "a" } " satisfying:" { $code "a*q = d mod p" } }
+{ $notes "GCD in the case of polynomials is a monic polynomial of the highest possible degree that divides into both " { $snippet "p" } " and " { $snippet "q" } "." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 1 1 1} { 1 1 } pgcd swap . ." "{ 0 0 }\n{ 1 1 }" } } ;
+
+HELP: pdiff
+{ $values { "p" "a polynomial" } { "p'" "a polynomial" } }
+{ $description "Finds the derivative of " { $snippet "p" } "." } ;
+
+HELP: polyval
+{ $values { "p" "a polynomial" } { "x" number } { "p[x]" number } }
+{ $description "Evaluate " { $snippet "p" } " with the input " { $snippet "x" } "." }
+{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 0 1 } 2 polyval ." "5" } } ;
+
diff --git a/extra/math/polynomials/polynomials-tests.factor b/extra/math/polynomials/polynomials-tests.factor
index cccf24fbff..cd88d19d13 100644
--- a/extra/math/polynomials/polynomials-tests.factor
+++ b/extra/math/polynomials/polynomials-tests.factor
@@ -1,7 +1,6 @@
-IN: math.polynomials.tests
 USING: kernel math math.polynomials tools.test ;
+IN: math.polynomials.tests
 
-! Tests
 [ { 0 1 } ] [ { 0 1 0 0 } ptrim ] unit-test
 [ { 1 } ] [ { 1 0 0 } ptrim ] unit-test
 [ { 0 } ] [ { 0 } ptrim ] unit-test
diff --git a/extra/math/polynomials/polynomials.factor b/extra/math/polynomials/polynomials.factor
index 47226114d0..13090b6486 100644
--- a/extra/math/polynomials/polynomials.factor
+++ b/extra/math/polynomials/polynomials.factor
@@ -4,46 +4,38 @@ USING: arrays kernel make math math.order math.vectors sequences shuffle
     splitting vectors ;
 IN: math.polynomials
 
-! Polynomials are vectors with the highest powers on the right:
-! { 1 1 0 1 } -> 1 + x + x^3
-! { } -> 0
-
-: powers ( n x -- seq )
-    #! Output sequence has n elements, { 1 x x^2 x^3 ... }
-    <array> 1 [ * ] accumulate nip ;
-
 <PRIVATE
 
-: 2pad-left ( p p n -- p p ) [ 0 pad-left ] curry bi@ ;
-: 2pad-right ( p p n -- p p ) [ 0 pad-right ] curry bi@ ;
-: pextend ( p p -- p p ) 2dup [ length ] bi@ max 2pad-right ;
-: pextend-left ( p p -- p p ) 2dup [ length ] bi@ max 2pad-left ;
+: 2pad-left ( p q n -- p q ) [ 0 pad-left ] curry bi@ ;
+: 2pad-right ( p q n -- p q ) [ 0 pad-right ] curry bi@ ;
+: pextend ( p q -- p q ) 2dup [ length ] bi@ max 2pad-right ;
+: pextend-left ( p q -- p q ) 2dup [ length ] bi@ max 2pad-left ;
 : unempty ( seq -- seq ) [ { 0 } ] when-empty ;
 : 2unempty ( seq seq -- seq seq ) [ unempty ] bi@ ;
 
 PRIVATE>
 
-: p= ( p p -- ? ) pextend = ;
+: powers ( n x -- seq )
+    <array> 1 [ * ] accumulate nip ;
+
+: p= ( p q -- ? ) pextend = ;
 
 : ptrim ( p -- p )
     dup length 1 = [ [ zero? ] trim-right ] unless ;
 
-: 2ptrim ( p p -- p p ) [ ptrim ] bi@ ;
-: p+ ( p p -- p ) pextend v+ ;
-: p- ( p p -- p ) pextend v- ;
+: 2ptrim ( p q -- p q ) [ ptrim ] bi@ ;
+: p+ ( p q -- r ) pextend v+ ;
+: p- ( p q -- r ) pextend v- ;
 : n*p ( n p -- n*p ) n*v ;
 
-! convolution
-: pextend-conv ( p p -- p p )
-    #! extend to: p_m + p_n - 1
+: pextend-conv ( p q -- p q )
     2dup [ length ] bi@ + 1- 2pad-right [ >vector ] bi@ ;
 
-: p* ( p p -- p )
-    #! Multiply two polynomials.
+: p* ( p q -- r )
     2unempty pextend-conv <reversed> dup length
     [ over length pick <slice> pick [ * ] 2map sum ] map 2nip reverse ;
 
-: p-sq ( p -- p-sq )
+: p-sq ( p -- p^2 )
     dup p* ;
 
 <PRIVATE
@@ -66,10 +58,12 @@ PRIVATE>
 
 PRIVATE>
 
-: p/mod ( a b -- / mod )
+: p/mod ( p q -- z w )
     p/mod-setup [ [ (p/mod) ] times ] V{ } make
     reverse nip swap 2ptrim pextend ;
 
+<PRIVATE
+
 : (pgcd) ( b a y x -- a d )
     dup V{ 0 } clone p= [
         drop nip
@@ -77,14 +71,14 @@ PRIVATE>
         tuck p/mod [ pick p* swap [ swapd p- ] dip ] dip (pgcd)
     ] if ;
 
-: pgcd ( p p -- p q )
+PRIVATE>
+
+: pgcd ( p q -- a d )
     swap V{ 0 } clone V{ 1 } clone 2swap (pgcd) [ >array ] bi@ ;
 
 : pdiff ( p -- p' )
-    #! Polynomial derivative.
     dup length v* { 0 } ?head drop ;
 
 : polyval ( p x -- p[x] )
-    #! Evaluate a polynomial.
     [ dup length ] dip powers v. ;