coefficient x14

Factorization of the Coefficient of x 14 Sean Trendell 10/16/2017 var ( 'x , a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 , p1 , p2 , p3 , p4 , q1 , q2 , q3 , q4 ' ) w0 = SR . w i l d ( 0 ) s a g e _ s e r v e r .MAX_STDOUT_SIZE=250000 s a g e _ s e r v e r .MAX_OUTPUT=250000 #This d e f i n e s our v a r i a b l e s , s e t s up our w i l d c a r d and i n c r e a s e s max output f o r \ t h e program . (x, a1, a2, a3, a4, b1, b2, b3, b4, p1, p2, p3, p4, q1, q2, q3, q4) E=(x - a1 * a2 * a3 * a4 ) * ( x - a1 * a2 * a3 *b4 ) * ( x - a1 * a2 *b3* a4 ) * ( x - a1 * a2 *b3*b4 ) * ( x - a1 *b2* a3 *\ a4 ) * ( x - a1 *b2* a3 *b4 ) * ( x - a1 *b2*b3* a4 ) * ( x - a1 *b2*b3*b4 ) * ( x - b1* a2 * a3 * a4 ) * ( x - b1* a2\ * a3 *b4 ) * ( x - b1* a2 *b3* a4 ) * ( x - b1* a2 *b3*b4 ) * ( x - b1*b2* a3 * a4 ) * ( x - b1*b2* a3 *b4 ) * ( x - \ b1*b2*b3* a4 ) * ( x - b1*b2*b3*b4 ) F=E . expand ( ) #M u l t i p l i c a t i o n o f t h e c h a r a c t e r i s t i c f u n c t i o n , t h i s i s not show do t o l e n g t h . f 1 4=F . c o e f f i c i e n t ( x , 1 4 ) f14 #Ga th er in g o f t h e c o e f f i c i e n t f o r x^14 from our e x p a n s i o n o f t h e c h a r a c t e r i s t i c \ f u n c t i o n . The g e n e r a l way we work t h i s out i s f i r s t s e p a r a t e t h e \ c o e f f i c i e n t by t h e n u m e r i c a l p a r t o f each term then s e p a r a t e i t f u r t h e r i n t o \ manageable gr ou p s t h a t have c l e a r s u b s t i t u t i o n . a1*a2^2*a3^2*a4^2*b1 + a1^2*a2*a3^2*a4^2*b2 + 2*a1*a2*a3^2*a4^2*b1*b2 + a2*a3^2*a4^2*b1^2*b2 + a1*a3^2*a4^2*b1*b2^2 + a1^2*a2^2*a3*a4^2*b3 + 2*a1*a2^2*a3*a4^2*b1*b3 + a2^2*a3*a4^2*b1^2*b3 + 2*a1^2*a2*a3*a4^2*b2*b3 + 4*a1*a2*a3*a4^2*b1*b2*b3 + 2*a2*a3*a4^2*b1^2*b2*b3 + a1^2*a3*a4^2*b2^2*b3 + 2*a1*a3*a4^2*b1*b2^2*b3 + a3*a4^2*b1^2*b2^2*b3 + a1*a2^2*a4^2*b1*b3^2 + a1^2*a2*a4^2*b2*b3^2 + 2*a1*a2*a4^2*b1*b2*b3^2 + a2*a4^2*b1^2*b2*b3^2 + a1*a4^2*b1*b2^2*b3^2 + a1^2*a2^2*a3^2*a4*b4 + 2*a1*a2^2*a3^2*a4*b1*b4 + a2^2*a3^2*a4*b1^2*b4 + 2*a1^2*a2*a3^2*a4*b2*b4 + 4*a1*a2*a3^2*a4*b1*b2*b4 + 2*a2*a3^2*a4*b1^2*b2*b4 + a1^2*a3^2*a4*b2^2*b4 + 2*a1*a3^2*a4*b1*b2^2*b4 + a3^2*a4*b1^2*b2^2*b4 + 2*a1^2*a2^2*a3*a4*b3*b4 + 4*a1*a2^2*a3*a4*b1*b3*b4 + 2*a2^2*a3*a4*b1^2*b3*b4 + 4*a1^2*a2*a3*a4*b2*b3*b4 + 8*a1*a2*a3*a4*b1*b2*b3*b4 + 4*a2*a3*a4*b1^2*b2*b3*b4 + 2*a1^2*a3*a4*b2^2*b3*b4 + 4*a1*a3*a4*b1*b2^2*b3*b4 + 2*a3*a4*b1^2*b2^2*b3*b4 + a1^2*a2^2*a4*b3^2*b4 + 2*a1*a2^2*a4*b1*b3^2*b4 + a2^2*a4*b1^2*b3^2*b4 + 2*a1^2*a2*a4*b2*b3^2*b4 + 4*a1*a2*a4*b1*b2*b3^2*b4 + 2*a2*a4*b1^2*b2*b3^2*b4 + a1^2*a4*b2^2*b3^2*b4 + 2*a1*a4*b1*b2^2*b3^2*b4 + a4*b1^2*b2^2*b3^2*b4 + a1*a2^2*a3^2*b1*b4^2 + a1^2*a2*a3^2*b2*b4^2 + 2*a1*a2*a3^2*b1*b2*b4^2 + a2*a3^2*b1^2*b2*b4^2 + a1*a3^2*b1*b2^2*b4^2 + a1^2*a2^2*a3*b3*b4^2 + 2*a1*a2^2*a3*b1*b3*b4^2 + a2^2*a3*b1^2*b3*b4^2 + 2*a1^2*a2*a3*b2*b3*b4^2 + 4*a1*a2*a3*b1*b2*b3*b4^2 + 2*a2*a3*b1^2*b2*b3*b4^2 + a1^2*a3*b2^2*b3*b4^2 + 2*a1*a3*b1*b2^2*b3*b4^2 + a3*b1^2*b2^2*b3*b4^2 + a1*a2^2*b1*b3^2*b4^2 + a1^2*a2*b2*b3^2*b4^2 + 2*a1*a2*b1*b2*b3^2*b4^2 + a2*b1^2*b2*b3^2*b4^2 + a1*b1*b2^2*b3^2*b4^2 c2=sum ( f 1 4 . f i n d ( 2 *w0) ) c2 #This f i n d s a l l our terms t h a t have a 2 i n them and then adds them t o g e t h e r . \ For t h i s sum we group terms such t h a t we have e x a c t l y two o f t h e a v a r i a b l e s \ 1  t o t h e f i r s t power . 2*a1*a2*a3^2*a4^2*b1*b2 + 2*a1*a2^2*a3*a4^2*b1*b3 + 2*a1^2*a2*a3*a4^2*b2*b3 + 2*a2*a3*a4^2*b1^2*b2*b3 + 2*a1*a3*a4^2*b1*b2^2*b3 + 2*a1*a2*a4^2*b1*b2*b3^2 + 2*a1*a2^2*a3^2*a4*b1*b4 + 2*a1^2*a2*a3^2*a4*b2*b4 + 2*a2*a3^2*a4*b1^2*b2*b4 + 2*a1*a3^2*a4*b1*b2^2*b4 + 2*a1^2*a2^2*a3*a4*b3*b4 + 2*a2^2*a3*a4*b1^2*b3*b4 + 2*a1^2*a3*a4*b2^2*b3*b4 + 2*a3*a4*b1^2*b2^2*b3*b4 + 2*a1*a2^2*a4*b1*b3^2*b4 + 2*a1^2*a2*a4*b2*b3^2*b4 + 2*a2*a4*b1^2*b2*b3^2*b4 + 2*a1*a4*b1*b2^2*b3^2*b4 + 2*a1*a2*a3^2*b1*b2*b4^2 + 2*a1*a2^2*a3*b1*b3*b4^2 + 2*a1^2*a2*a3*b2*b3*b4^2 + 2*a2*a3*b1^2*b2*b3*b4^2 + 2*a1*a3*b1*b2^2*b3*b4^2 + 2*a1*a2*b1*b2*b3^2*b4^2 c2a1a2=sum ( c2 . f i n d ( a1 * a2 *w0) ) c2a1a2 c2a1a2 . f a c t o r ( ) s2 a 1 a 2 =2*(p3^2+2*q3 ) * ( p4^2+2*q4 ) * q1 * q2 s2 a 1 a 2 #This i s t h e sum o f terms with a1 and a2 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . 2*a1*a2*a3^2*a4^2*b1*b2 + 2*a1*a2*a4^2*b1*b2*b3^2 + 2*a1*a2*a3^2*b1*b2*b4^2 + 2*a1*a2*b1*b2*b3^2*b4^2 2*(a3^2 + b3^2)*(a4^2 + b4^2)*a1*a2*b1*b2 2*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1*q2 c2a1a3=sum ( c2 . f i n d ( a1 * a3 *w0) ) c2a1a3 c2a1a3 . f a c t o r ( ) s2 a 1 a 3 =2*(p2^2+2*q2 ) * ( p4^2+2*q4 ) * q1 * q3 s2 a 1 a 3 #This i s t h e sum o f terms with a1 and a3 i n them and , i t s f a c t o r i z a t i o n , and \ its substitution . 2*a1*a2^2*a3*a4^2*b1*b3 + 2*a1*a3*a4^2*b1*b2^2*b3 + 2*a1*a2^2*a3*b1*b3*b4^2 + 2*a1*a3*b1*b2^2*b3*b4^2 2*(a2^2 + b2^2)*(a4^2 + b4^2)*a1*a3*b1*b3 2*(p2^2 + 2*q2)*(p4^2 + 2*q4)*q1*q3 c2a1a4=sum ( c2 . f i n d ( a1 * a4 *w0) ) c2a1a4 c2a1a4 . f a c t o r ( ) s2 a 1 a 4 =2*(p2^2+2*q2 ) * ( p3^2+2*q3 ) * q1 * q4 s2 a 1 a 4 #This i s t h e sum o f terms with a1 and a4 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . 2*a1*a2^2*a3^2*a4*b1*b4 + 2*a1*a3^2*a4*b1*b2^2*b4 + 2*a1*a2^2*a4*b1*b3^2*b4 + 2*a1*a4*b1*b2^2*b3^2*b4 2*(a2^2 + b2^2)*(a3^2 + b3^2)*a1*a4*b1*b4 2*(p2^2 + 2*q2)*(p3^2 + 2*q3)*q1*q4 c2a2a3=sum ( c2 . f i n d ( a2 * a3 *w0) ) c2a2a3 c2a2a3 . f a c t o r ( ) s2 a 2 a 3 =2*(p1^2+2*q1 ) * ( p4^2+2*q4 ) * q2 * q3 s2 a 2 a 3 #This i s t h e sum o f terms with a2 and a3 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . 2*a1^2*a2*a3*a4^2*b2*b3 + 2*a2*a3*a4^2*b1^2*b2*b3 + 2*a1^2*a2*a3*b2*b3*b4^2 + 2*a2*a3*b1^2*b2*b3*b4^2 2*(a1^2 + b1^2)*(a4^2 + b4^2)*a2*a3*b2*b3 2*(p1^2 + 2*q1)*(p4^2 + 2*q4)*q2*q3 c2a2a4=sum ( c2 . f i n d ( a2 * a4 *w0) ) c2a2a4 c2a2a4 . f a c t o r ( ) 2  s2 a 2 a 4 =2*(p1^2+2*q1 ) * ( p3^2+2*q3 ) * q2 * q4 s2 a 2 a 4 #This i s t h e sum o f terms with a2 and a4 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . 2*a1^2*a2*a3^2*a4*b2*b4 + 2*a2*a3^2*a4*b1^2*b2*b4 + 2*a1^2*a2*a4*b2*b3^2*b4 + 2*a2*a4*b1^2*b2*b3^2*b4 2*(a1^2 + b1^2)*(a3^2 + b3^2)*a2*a4*b2*b4 2*(p1^2 + 2*q1)*(p3^2 + 2*q3)*q2*q4 c2a3a4=sum ( c2 . f i n d ( a3 * a4 *w0) ) c2a3a4 c2a3a4 . f a c t o r ( ) s2 a 3 a 4 =2*(p1^2+2*q1 ) * ( p2^2+2*q2 ) * q3 * q4 s2 a 3 a 4 #This i s t h e sum o f terms with a3 and a4 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . 2*a1^2*a2^2*a3*a4*b3*b4 + 2*a2^2*a3*a4*b1^2*b3*b4 + 2*a1^2*a3*a4*b2^2*b3*b4 + 2*a3*a4*b1^2*b2^2*b3*b4 2*(a1^2 + b1^2)*(a2^2 + b2^2)*a3*a4*b3*b4 2*(p1^2 + 2*q1)*(p2^2 + 2*q2)*q3*q4 c4=sum ( f 1 4 . f i n d ( 4 *w0) ) c4 #This f i n d s a l l our terms t h a t have a 4 i n them and then adds them t o g e t h e r . \ For t h i s sum we f a c t o r such t h a t each group has t h e term with one o f our a \ v a r i a b l e s and i t s c o r r e s p o n d i n g b both t o t h e s e c o n d power . 4*a1*a2*a3*a4^2*b1*b2*b3 + 4*a1*a2*a3^2*a4*b1*b2*b4 + 4*a1*a2^2*a3*a4*b1*b3*b4 + 4*a1^2*a2*a3*a4*b2*b3*b4 + 4*a2*a3*a4*b1^2*b2*b3*b4 + 4*a1*a3*a4*b1*b2^2*b3*b4 + 4*a1*a2*a4*b1*b2*b3^2*b4 + 4*a1*a2*a3*b1*b2*b3*b4^2 c4a1=sum ( c4 . f i n d ( a1 ^2*w0) )+sum ( c4 . f i n d ( b1 ^2*w0) ) c4a1 c4a1 . f a c t o r ( ) s 4 a 1 = -4*( p1^2+2*q1 ) * q2 * q3 * q4 s4a1 #This i s t h e sum o f t h e term with a1 ^2 and t h e term with b1 ^2 , i t s \ f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n . 4*a1^2*a2*a3*a4*b2*b3*b4 + 4*a2*a3*a4*b1^2*b2*b3*b4 4*(a1^2 + b1^2)*a2*a3*a4*b2*b3*b4 -4*(p1^2 + 2*q1)*q2*q3*q4 c4a2=sum ( c4 . f i n d ( a2 ^2*w0) )+sum ( c4 . f i n d ( b2 ^2*w0) ) c4a2 c4a2 . f a c t o r ( ) s 4 a 2 = -4*( p2^2+2*q2 ) * q1 * q3 * q4 s4a2 #This i s t h e sum o f t h e term with a2 ^2 and t h e term with b2 ^2 , i t s \ f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n . 4*a1*a2^2*a3*a4*b1*b3*b4 + 4*a1*a3*a4*b1*b2^2*b3*b4 4*(a2^2 + b2^2)*a1*a3*a4*b1*b3*b4 -4*(p2^2 + 2*q2)*q1*q3*q4 c4a3=sum ( c4 . f i n d ( a3 ^2*w0) )+sum ( c4 . f i n d ( b3 ^2*w0) ) c4a3 c4a3 . f a c t o r ( ) s 4 a 3 = -4*( p3^2+2*q3 ) * q1 * q2 * q4 s4a3 #This i s t h e sum o f t h e term with a3 ^2 and t h e term with b3 ^2 , i t s \ f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n . 4*a1*a2*a3^2*a4*b1*b2*b4 + 4*a1*a2*a4*b1*b2*b3^2*b4 4*(a3^2 + b3^2)*a1*a2*a4*b1*b2*b4 3  -4*(p3^2 + 2*q3)*q1*q2*q4 c4a4=sum ( c4 . f i n d ( a4 ^2*w0) )+sum ( c4 . f i n d ( b4 ^2*w0) ) c4a4 c4a4 . f a c t o r ( ) s 4 a 4 = -4*( p4^2+2*q4 ) * q1 * q2 * q3 s4a4 #This i s t h e sum o f t h e term with a4 ^2 and t h e term with b4 ^2 , i t s \ f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n . 4*a1*a2*a3*a4^2*b1*b2*b3 + 4*a1*a2*a3*b1*b2*b3*b4^2 4*(a4^2 + b4^2)*a1*a2*a3*b1*b2*b3 -4*(p4^2 + 2*q4)*q1*q2*q3 c8=sum ( f 1 4 . f i n d ( 8 *w0) ) c8 s 8 =8*q1 * q2 * q3 * q4 s8 #This f i n d s our term t h a t has a 8 i n i t and i t s s u b s t i t u t i o n . 8*a1*a2*a3*a4*b1*b2*b3*b4 8*q1*q2*q3*q4 c1=sum ( f 1 4 . f i n d ( 1 *w0) ) - c2 - c4 - c8 c1 #This f i n d s a l l our terms t h a t do not have a number o t h e r t h e r then 1 i n them \ and then adds them t o g e t h e r . For t h i s sum we group terms such t h a t we have \ e x a c t l y one o f t h e a v a r i a b l e s t o t h e f i r s t power . a1*a2^2*a3^2*a4^2*b1 + a1^2*a2*a3^2*a4^2*b2 + a2*a3^2*a4^2*b1^2*b2 + a1*a3^2*a4^2*b1*b2^2 + a1^2*a2^2*a3*a4^2*b3 + a2^2*a3*a4^2*b1^2*b3 + a1^2*a3*a4^2*b2^2*b3 + a3*a4^2*b1^2*b2^2*b3 + a1*a2^2*a4^2*b1*b3^2 + a1^2*a2*a4^2*b2*b3^2 + a2*a4^2*b1^2*b2*b3^2 + a1*a4^2*b1*b2^2*b3^2 + a1^2*a2^2*a3^2*a4*b4 + a2^2*a3^2*a4*b1^2*b4 + a1^2*a3^2*a4*b2^2*b4 + a3^2*a4*b1^2*b2^2*b4 + a1^2*a2^2*a4*b3^2*b4 + a2^2*a4*b1^2*b3^2*b4 + a1^2*a4*b2^2*b3^2*b4 + a4*b1^2*b2^2*b3^2*b4 + a1*a2^2*a3^2*b1*b4^2 + a1^2*a2*a3^2*b2*b4^2 + a2*a3^2*b1^2*b2*b4^2 + a1*a3^2*b1*b2^2*b4^2 + a1^2*a2^2*a3*b3*b4^2 + a2^2*a3*b1^2*b3*b4^2 + a1^2*a3*b2^2*b3*b4^2 + a3*b1^2*b2^2*b3*b4^2 + a1*a2^2*b1*b3^2*b4^2 + a1^2*a2*b2*b3^2*b4^2 + a2*b1^2*b2*b3^2*b4^2 + a1*b1*b2^2*b3^2*b4^2 c1a1=sum ( c1 . f i n d ( a1 *w0) ) c1a1 c1a1 . f a c t o r ( ) s 1 a 1 = -( p2^2 + 2* q2 ) * ( p3^2 + 2* q3 ) * ( p4^2 + 2* q4 ) * q1 s1a1 #This i s t h e sum o f t h e term with a1 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . a1*a2^2*a3^2*a4^2*b1 + a1*a3^2*a4^2*b1*b2^2 + a1*a2^2*a4^2*b1*b3^2 + a1*a4^2*b1*b2^2*b3^2 + a1*a2^2*a3^2*b1*b4^2 + a1*a3^2*b1*b2^2*b4^2 + a1*a2^2*b1*b3^2*b4^2 + a1*b1*b2^2*b3^2*b4^2 (a2^2 + b2^2)*(a3^2 + b3^2)*(a4^2 + b4^2)*a1*b1 -(p2^2 + 2*q2)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1 c1a2=sum ( c1 . f i n d ( a2 *w0) ) c1a2 c1a2 . f a c t o r ( ) s 1 a 2 = -( p1^2 + 2* q1 ) * ( p3^2 + 2* q3 ) * ( p4^2 + 2* q4 ) * q2 s1a2 #This i s t h e sum o f t h e term with a2 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . a1^2*a2*a3^2*a4^2*b2 + a2*a3^2*a4^2*b1^2*b2 + a1^2*a2*a4^2*b2*b3^2 + a2*a4^2*b1^2*b2*b3^2 + a1^2*a2*a3^2*b2*b4^2 + a2*a3^2*b1^2*b2*b4^2 + a1^2*a2*b2*b3^2*b4^2 + a2*b1^2*b2*b3^2*b4^2 (a1^2 + b1^2)*(a3^2 + b3^2)*(a4^2 + b4^2)*a2*b2 -(p1^2 + 2*q1)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q2 4  c1a3=sum ( c1 . f i n d ( a3 *w0) ) c1a3 c1a3 . f a c t o r ( ) s 1 a 3 = -( p1^2 + 2* q1 ) * ( p2^2 + 2* q2 ) * ( p4^2 + 2* q4 ) * q3 s1a3 #This i s t h e sum o f t h e term with a3 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . a1^2*a2^2*a3*a4^2*b3 + a2^2*a3*a4^2*b1^2*b3 + a1^2*a3*a4^2*b2^2*b3 + a3*a4^2*b1^2*b2^2*b3 + a1^2*a2^2*a3*b3*b4^2 + a2^2*a3*b1^2*b3*b4^2 + a1^2*a3*b2^2*b3*b4^2 + a3*b1^2*b2^2*b3*b4^2 (a1^2 + b1^2)*(a2^2 + b2^2)*(a4^2 + b4^2)*a3*b3 -(p1^2 + 2*q1)*(p2^2 + 2*q2)*(p4^2 + 2*q4)*q3 c1a4=sum ( c1 . f i n d ( a4 *w0) ) c1a4 c1a4 . f a c t o r ( ) s 1 a 4 = -( p1^2 + 2* q1 ) * ( p2^2 + 2* q2 ) * ( p3^2 + 2* q3 ) * q4 s1a4 #This i s t h e sum o f t h e term with a4 i n them , i t s f a c t o r i z a t i o n , and i t s \ substitution . a1^2*a2^2*a3^2*a4*b4 + a2^2*a3^2*a4*b1^2*b4 + a1^2*a3^2*a4*b2^2*b4 + a3^2*a4*b1^2*b2^2*b4 + a1^2*a2^2*a4*b3^2*b4 + a2^2*a4*b1^2*b3^2*b4 + a1^2*a4*b2^2*b3^2*b4 + a4*b1^2*b2^2*b3^2*b4 (a1^2 + b1^2)*(a2^2 + b2^2)*(a3^2 + b3^2)*a4*b4 -(p1^2 + 2*q1)*(p2^2 + 2*q2)*(p3^2 + 2*q3)*q4 L=s 1 a 1+s 1 a 2+s 1 a 3+s 1 a 4+s 2 a1 a 2+s 2 a1 a 3+s 2 a 1 a 4+s 2 a 2 a 3+s 2 a 2 a 4+s 2 a 3 a 4+s 4 a 1+s 4 a 2+s 4 a 3+\ s 4 a 4+s 8 L #This l i n e b r i n g s a l l t h e group s u b s t i t u t i o n s back i n t o one e x p r e s s i o n -(p2^2 + 2*q2)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1 - (p1^2 + 2*q1)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q2 + 2*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1*q2 - (p1^2 + 2*q1)*(p2^2 + 2*q2)*(p4^2 + 2*q4)*q3 + 2*(p2^2 + 2*q2)*(p4^2 + 2*q4)*q1*q3 + 2*(p1^2 + 2*q1)*(p4^2 + 2*q4)*q2*q3 4*(p4^2 + 2*q4)*q1*q2*q3 - (p1^2 + 2*q1)*(p2^2 + 2*q2)*(p3^2 + 2*q3)*q4 + 2*(p2^2 + 2*q2)*(p3^2 + 2*q3)*q1*q4 + 2*(p1^2 + 2*q1)*(p3^2 + 2*q3)*q2*q4 - 4*(p3^2 + 2*q3)*q1*q2*q4 + 2*(p1^2 + 2*q1)*(p2^2 + 2*q2)*q3*q4 - 4*(p2^2 + 2*q2)*q1*q3*q4 - 4*(p1^2 + 2*q1)*q2*q3*q4 + 8*q1*q2*q3*q4 L1=L( p1=a1+b1 , p2=a2+b2 , p3=a3+b3 , p4=a4+b4 , q1 =-1* a1 *b1 , q2 =-1* a2 *b2 , q3 =-1* a3\ *b3 , q4 =-1* a4 *b4 ) b o o l ( L1==f 1 4 ) #This i s a check t o make s u r e t h a t when back s u b s t i t u t e we end up with t h e \ original coefficient . True L . expand ( ) l a t e x ( - L . expand ( ) ) #This i s t h e e x p a n s i o n o f our c o e f f i c i e n t and a l a t e x output f o r e a s y c o p y i n g \ t o t h e paper . -p2^2*p3^2*p4^2*q1 - p1^2*p3^2*p4^2*q2 - p1^2*p2^2*p4^2*q3 - p1^2*p2^2*p3^2*q4 2*p3^2*p4^2*q1*q2 - 2*p2^2*p4^2*q1*q3 - 2*p1^2*p4^2*q2*q3 - 2*p2^2*p3^2*q1*q4 2*p1^2*p3^2*q2*q4 - 2*p1^2*p2^2*q3*q4 - 4*p4^2*q1*q2*q3 - 4*p3^2*q1*q2*q4 4*p2^2*q1*q3*q4 - 4*p1^2*q2*q3*q4 - 8*q1*q2*q3*q4 p_{2}^{2} p_{3}^{2} p_{4}^{2} q_{1} + p_{1}^{2} p_{3}^{2} p_{4}^{2} q_{2} + p_{1}^{2} p_{2}^{2} p_{4}^{2} q_{3} + p_{1}^{2} p_{2}^{2} p_{3}^{2} q_{4} + 2 \, p_{3}^{2} p_{4}^{2} q_{1} q_{2} + 2 \, p_{2}^{2} p_{4}^{2} q_{1} q_{3} + 2 \, p_{1}^{2} p_{4}^{2} q_{2} q_{3} + 2 \, p_{2}^{2} p_{3}^{2} q_{1} q_{4} + 2 \, p_{1}^{2} p_{3}^{2} q_{2} q_{4} + 2 \, p_{1}^{2} p_{2}^{2} q_{3} q_{4} + 4 \, p_{4}^{2} q_{1} q_{2} q_{3} + 4 \, p_{3}^{2} q_{1} q_{2} q_{4} + 4 \, p_{2}^{2} q_{1} q_{3} q_{4} + 4 \, p_{1}^{2} q_{2} q_{3} q_{4} + 8 \, q_{1} q_{2} q_{3} q_{4} 5