Question on alternate forms of polynomial output

Question

EDIT: Would it be possible to do something like let $y=ax+b$ then use Collect[] or Apart[] on the new expression? How would I go about this. I've tried using Collect[%,ax+b], Collect[%,{ax+b}], and the same for Apart[] but it doens't work.

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I have equations such as this:

Latex: $225 a^4 + 600 a^3 b^2 + 520 a^2 b^4 + 160 a b^6 + 16 b^8 + 1200 a^4 b x + 2080 a^3 b^3 x + 960 a^2 b^5 x + 128 a b^7 x + 600 a^5 x^2 + 3120 a^4 b^2 x^2 + 2400 a^3 b^4 x^2 + 448 a^2 b^6 x^2 + 2080 a^5 b x^3 + 3200 a^4 b^3 x^3 + 896 a^3 b^5 x^3 + 520 a^6 x^4 + 2400 a^5 b^2 x^4 + 1120 a^4 b^4 x^4 + 960 a^6 b x^5 + 896 a^5 b^3 x^5 + 160 a^7 x^6 + 448 a^6 b^2 x^6 + 128 a^7 b x^7 + 16 a^8 x^8$

Mathematica: 225 a^4 + 600 a^3 b^2 + 520 a^2 b^4 + 160 a b^6 + 16 b^8 + 1200 a^4 b x + 2080 a^3 b^3 x + 960 a^2 b^5 x + 128 a b^7 x + 600 a^5 x^2 + 3120 a^4 b^2 x^2 + 2400 a^3 b^4 x^2 + 448 a^2 b^6 x^2 + 2080 a^5 b x^3 + 3200 a^4 b^3 x^3 + 896 a^3 b^5 x^3 + 520 a^6 x^4 + 2400 a^5 b^2 x^4 + 1120 a^4 b^4 x^4 + 960 a^6 b x^5 + 896 a^5 b^3 x^5 + 160 a^7 x^6 + 448 a^6 b^2 x^6 + 128 a^7 b x^7 + 16 a^8 x^8

which has an awesome simplified form that I did by hand as:

Latex: $(15a^2 + 20a(ax+b)^2 + 4(ax+b)^4)^2$

Mathematica: (15a^2 + 20a(ax+b)^2 + 4(ax+b)^4)^2

However, these get much more difficult when I deal with things like (Note: I have hundreds of expressions like this, so I need general help with the types of expressions to get things in an alternate form, not just this one done) :

Latex: $108056025 a^{10} + 720373500 a^9 b^2 + 1776921300 a^8 b^4 + 2085652800 a^7 b^6 + 1335549600 a^6 b^8 + 500734080 a^5 b^{10} + 113731200 a^4 b^{12} + 15713280 a^3 b^{14} + 1281280 a^2 b^{16} + 56320 a b^{18} + 1024 b^{20} + 1440747000 a^{10} b x + 7107685200 a^9 b^3 x + 12513916800 a^8 b^5 x + 10684396800 a^7 b^7 x + 5007340800 a^6 b^9 x + 1364774400 a^5 b^{11} x + 219985920 a^4 b^{13} x + 20500480 a^3 b^{15} x + 1013760 a^2 b^{17} x + 20480 a b^{19} x + 720373500 a^{11} x^2 + 10661527800 a^{10} b^2 x^2 + 31284792000 a^9 b^4 x^2 + 37395388800 a^8 b^6 x^2 + 22533033600 a^7 b^8 x^2 + 7506259200 a^6 b^{10} x^2 + 1429908480 a^5 b^{12} x^2 + 153753600 a^4 b^{14} x^2 + 8616960 a^3 b^{16} x^2 + 194560 a^2 b^{18} x^2 + 7107685200 a^{11} b x^3 + 41713056000 a^{10} b^3 x^3 + 74790777600 a^9 b^5 x^3 + 60088089600 a^8 b^7 x^3 + 25020864000 a^7 b^9 x^3 + 5719633920 a^6 b^{11} x^3 + 717516800 a^5 b^{13} x^3 + 45957120 a^4 b^{15} x^3 + 1167360 a^3 b^{17} x^3 + 1776921300 a^{12} x^4 + 31284792000 a^{11} b^2 x^4 + 93488472000 a^{10} b^4 x^4 + 105154156800 a^9 b^6 x^4 + 56296944000 a^8 b^8 x^4 + 15728993280 a^7 b^{10} x^4 + 2331929600 a^6 b^{12} x^4 + 172339200 a^5 b^{14} x^4 + 4961280 a^4 b^{16} x^4 + 12513916800 a^{12} b x^5 + 74790777600 a^{11} b^3 x^5 + 126184988160 a^{10} b^5 x^5 + 90075110400 a^9 b^7 x^5 + 31457986560 a^8 b^9 x^5 + 5596631040 a^7 b^{11} x^5 + 482549760 a^6 b^{13} x^5 + 15876096 a^5 b^{15} x^5 + 2085652800 a^{13} x^6 + 37395388800 a^{12} b^2 x^6 + 105154156800 a^{11} b^4 x^6 + 105087628800 a^{10} b^6 x^6 + 47186979840 a^9 b^8 x^6 + 10260490240 a^8 b^{10} x^6 + 1045524480 a^7 b^{12} x^6 + 39690240 a^6 b^{14} x^6 + 10684396800 a^{13} b x^7 + 60088089600 a^{12} b^3 x^7 + 90075110400 a^{11} b^5 x^7 + 53927976960 a^{10} b^7 x^7 + 14657843200 a^9 b^9 x^7 + 1792327680 a^8 b^{11} x^7 + 79380480 a^7 b^{13} x^7 + 1335549600 a^{14} x^8 + 22533033600 a^{13} b^2 x^8 + 56296944000 a^{12} b^4 x^8 + 47186979840 a^{11} b^6 x^8 + 16490073600 a^{10} b^8 x^8 + 2464450560 a^9 b^{10} x^8 + 128993280 a^8 b^{12} x^8 + 5007340800 a^{14} b x^9 + 25020864000 a^{13} b^3 x^9 + 31457986560 a^{12} b^5 x^9 + 14657843200 a^{11} b^7 x^9 + 2738278400 a^{10} b^9 x^9 + 171991040 a^9 b^{11} x^9 + 500734080 a^{15} x^{10} + 7506259200 a^{14} b^2 x^{10} + 15728993280 a^{13} b^4 x^{10} + 10260490240 a^{12} b^6 x^{10} + 2464450560 a^{11} b^8 x^{10} + 189190144 a^{10} b^{10} x^{10} + 1364774400 a^{15} b^3 x^{11} + 5719633920 a^{14} b^3 x^{11} + 5596631040 a^{13} b^5 x^{11} + 1792327680 a^{12} b^7 x^{11} + 171991040 a^{11} b^9 x^{11} + 113731200 a^{16} x^{12} + 1429908480 a^{15} b^2 x^{12} + 2331929600 a^{14} b^4 x^{12} + 1045524480 a^{13} b^6 x^{12} + 128993280 a^{12} b^8 x^{12} + 219985920 a^{16} b x^{13} + 717516800 a^{15} b^3 x^{13} + 482549760 a^{14} b^5 x^{13} + 79380480 a^{13} b^7 x^{13} + 15713280 a^{17} x^{14} + 153753600 a^{16} b^2 x^{14} + 172339200 a^{15} b^4 x^{14} + 39690240 a^{14} b^6 x^{14} + 20500480 a^{17} b x^{15} + 45957120 a^{16} b^3 x^{15} + 15876096 a^{15} b^5 x^{15} + 1281280 a^{18} x^{16} + 8616960 a^{17} b^2 x^{16} + 4961280 a^{16} b^4 x^{16} + 1013760 a^{18} b x^{17} + 1167360 a^{17} b^3 x^{17} + 56320 a^{19} x^{18} + 194560 a^{18} b^2 x^{18} + 20480 a^{19} b x^{19} + 1024 a^{20} x^{20}$

Mathematica: 108056025 a^10 + 720373500 a^9 b^2 + 1776921300 a^8 b^4 + 2085652800 a^7 b^6 + 1335549600 a^6 b^8 + 500734080 a^5 b^10 + 113731200 a^4 b^12 + 15713280 a^3 b^14 + 1281280 a^2 b^16 + 56320 a b^18 + 1024 b^20 + 1440747000 a^10 b x + 7107685200 a^9 b^3 x + 12513916800 a^8 b^5 x + 10684396800 a^7 b^7 x + 5007340800 a^6 b^9 x + 1364774400 a^5 b^11 x + 219985920 a^4 b^13 x + 20500480 a^3 b^15 x + 1013760 a^2 b^17 x + 20480 a b^19 x + 720373500 a^11 x^2 + 10661527800 a^10 b^2 x^2 + 31284792000 a^9 b^4 x^2 + 37395388800 a^8 b^6 x^2 + 22533033600 a^7 b^8 x^2 + 7506259200 a^6 b^10 x^2 + 1429908480 a^5 b^12 x^2 + 153753600 a^4 b^14 x^2 + 8616960 a^3 b^16 x^2 + 194560 a^2 b^18 x^2 + 7107685200 a^11 b x^3 + 41713056000 a^10 b^3 x^3 + 74790777600 a^9 b^5 x^3 + 60088089600 a^8 b^7 x^3 + 25020864000 a^7 b^9 x^3 + 5719633920 a^6 b^11 x^3 + 717516800 a^5 b^13 x^3 + 45957120 a^4 b^15 x^3 + 1167360 a^3 b^17 x^3 + 1776921300 a^12 x^4 + 31284792000 a^11 b^2 x^4 + 93488472000 a^10 b^4 x^4 + 105154156800 a^9 b^6 x^4 + 56296944000 a^8 b^8 x^4 + 15728993280 a^7 b^10 x^4 + 2331929600 a^6 b^12 x^4 + 172339200 a^5 b^14 x^4 + 4961280 a^4 b^16 x^4 + 12513916800 a^12 b x^5 + 74790777600 a^11 b^3 x^5 + 126184988160 a^10 b^5 x^5 + 90075110400 a^9 b^7 x^5 + 31457986560 a^8 b^9 x^5 + 5596631040 a^7 b^11 x^5 + 482549760 a^6 b^13 x^5 + 15876096 a^5 b^15 x^5 + 2085652800 a^13 x^6 + 37395388800 a^12 b^2 x^6 + 105154156800 a^11 b^4 x^6 + 105087628800 a^10 b^6 x^6 + 47186979840 a^9 b^8 x^6 + 10260490240 a^8 b^10 x^6 + 1045524480 a^7 b^12 x^6 + 39690240 a^6 b^14 x^6 + 10684396800 a^13 b x^7 + 60088089600 a^12 b^3 x^7 + 90075110400 a^11 b^5 x^7 + 53927976960 a^10 b^7 x^7 + 14657843200 a^9 b^9 x^7 + 1792327680 a^8 b^11 x^7 + 79380480 a^7 b^13 x^7 + 1335549600 a^14 x^8 + 22533033600 a^13 b^2 x^8 + 56296944000 a^12 b^4 x^8 + 47186979840 a^11 b^6 x^8 + 16490073600 a^10 b^8 x^8 + 2464450560 a^9 b^10 x^8 + 128993280 a^8 b^12 x^8 + 5007340800 a^14 b x^9 + 25020864000 a^13 b^3 x^9 + 31457986560 a^12 b^5 x^9 + 14657843200 a^11 b^7 x^9 + 2738278400 a^10 b^9 x^9 + 171991040 a^9 b^11 x^9 + 500734080 a^15 x^10 + 7506259200 a^14 b^2 x^10 + 15728993280 a^13 b^4 x^10 + 10260490240 a^12 b^6 x^10 + 2464450560 a^11 b^8 x^10 + 189190144 a^10 b^10 x^10 + 5719633920 a^14 b^3 x^11 + 1364774400 a^15 b^3 x^11 + 5596631040 a^13 b^5 x^11 + 1792327680 a^12 b^7 x^11 + 171991040 a^11 b^9 x^11 + 113731200 a^16 x^12 + 1429908480 a^15 b^2 x^12 + 2331929600 a^14 b^4 x^12 + 1045524480 a^13 b^6 x^12 + 128993280 a^12 b^8 x^12 + 219985920 a^16 b x^13 + 717516800 a^15 b^3 x^13 + 482549760 a^14 b^5 x^13 + 79380480 a^13 b^7 x^13 + 15713280 a^17 x^14 + 153753600 a^16 b^2 x^14 + 172339200 a^15 b^4 x^14 + 39690240 a^14 b^6 x^14 + 20500480 a^17 b x^15 + 45957120 a^16 b^3 x^15 + 15876096 a^15 b^5 x^15 + 1281280 a^18 x^16 + 8616960 a^17 b^2 x^16 + 4961280 a^16 b^4 x^16 + 1013760 a^18 b x^17 + 1167360 a^17 b^3 x^17 + 56320 a^19 x^18 + 194560 a^18 b^2 x^18 + 20480 a^19 b x^19 + 1024 a^20 x^20

Where full simplifying gives this:

1024 b^20 + 1024 a^20 x^20 + 5120 a b^18 (11 + 4 b x) + 5120 a^19 x^18 (11 + 4 b x) + 1280 a^2 b^16 (1001 + 8 b x (99 + 19 b x)) + 1280 a^18 x^16 (1001 + 8 b x (99 + 19 b x)) + 5120 a^3 b^14 (3069 + b x (4004 + 3 b x (561 + 76 b x))) + 5120 a^17 x^14 (3069 + b x (4004 + 3 b x (561 + 76 b x))) + 1920 a^4 b^12 (59235 + 8 b x (14322 + b x (10010 + 17 b x (176 + 19 b x)))) + 1920 a^16 x^12 (59235 + 8 b x (14322 + b x (10010 + 17 b x (176 + 19 b x)))) + 128 a^5 b^10 (3911985 + 4 b x (2665575 + 2 b x (1396395 + 4 b x (175175 + 51 b x (825 + 76 b x))))) + 128 a^15 x^10 (3911985 + 4 b^2 x (2792790 x + b (2665575 + 8 x^2 (175175 + 51 b x (825 + 76 b x))))) + 160 a^6 b^8 (8347185 + 8 b x (3911985 + b x (5864265 + 4 b x (1117116 + b x (455455 + 408 b x (231 + 19 b x)))))) + 160 a^14 x^8 (8347185 + 8 b x (3911985 + b x (5864265 + 4 b x (1117116 + b x (455455 + 408 b x (231 + 19 b x)))))) + 960 a^7 b^6 (2172555 + 2 b x (5564790 + b x (11735955 + 4 b x (3257925 + 2 b x (1024023 + 4 b x (91091 + 17 b x (1001 + 76 b x))))))) + 960 a^13 x^6 (2172555 + 2 b x (5564790 + b x (11735955 + 4 b x (3257925 + 2 b x (1024023 + 4 b x (91091 + 17 b x (1001 + 76 b x))))))) + 20 a^8 b^4 (88846065 + 32 b x (19552995 + b x (58430295 + b x (93887640 + b x (87963975 + 208 b x (236313 + b x (77077 + 51 b x (264 + 19 b x)))))))) + 20 a^12 x^4 (88846065 + 32 b x (19552995 + b x (58430295 + b x (93887640 + b x (87963975 + 208 b x (236313 + b x (77077 + 51 b x (264 + 19 b x)))))))) + 20 a^9 b^2 (36018675 + 4 b x (88846065 + 4 b x (97764975 + 4 b x (58430295 + b x (82151685 + 4 b x (17592795 + 13 b x (708939 + 2 b x (110110 + 17 b x (1089 + 76 b x))))))))) + 20 a^11 x^2 (36018675 + 4 b x (88846065 + 4 b x (97764975 + 4 b x (58430295 + b x (82151685 + 4 b x (17592795 + 13 b x (708939 + 2 b x (110110 + 17 b x (1089 + 76 b x))))))))) + 11 a^10 (9823275 + 8 b x (16372125 + b x (121153725 + 8 b x (59251500 + b x (132796125 + 8 b x (22405005 + b x (18659025 + 13 b x (736560 + b x (225225 + 136 b x (275 + 19 b x))))))))))

So as you can see I need a way to simplify it in a similar way. However, I'm having trouble finding a way to get mathematica to simplify it in this way. Any suggestions? Also is there a list of all the ways I can simplify expressions? I have been going to the mathematica site and just looking at similar expressions to "simplify", "expand", "part", "apart", etc...

Answer 1

e1 = 225 a^4 + 600 a^3 b^2 + 520 a^2 b^4 + 160 a b^6 + 16 b^8 + 
   1200 a^4 b x + 2080 a^3 b^3 x + 960 a^2 b^5 x + 128 a b^7 x + 
   600 a^5 x^2 + 3120 a^4 b^2 x^2 + 2400 a^3 b^4 x^2 + 
   448 a^2 b^6 x^2 + 2080 a^5 b x^3 + 3200 a^4 b^3 x^3 + 
   896 a^3 b^5 x^3 + 520 a^6 x^4 + 2400 a^5 b^2 x^4 + 
   1120 a^4 b^4 x^4 + 960 a^6 b x^5 + 896 a^5 b^3 x^5 + 160 a^7 x^6 + 
   448 a^6 b^2 x^6 + 128 a^7 b x^7 + 16 a^8 x^8;

then

e1 /. x -> (y - b)/a // Expand // Factor

(15 a^2 + 20 a y^2 + 4 y^4)^2

Answer 2

FullSimplify[] on the enormous expression gives this answer in less than a second:

$(4 b^4 + 4 a^4 x^4 + 4 a b^2 (5 + 4 b x) + 4 a^3 x^2 (5 + 4 b x) + a^2 (15 + 8 b x (5 + 3 b x)))^2$

It would take me several days to calculate such a result "by hand," and no doubt I would make errors.

Answer 3

This is not a robust solution but since no one else was motivated to try to match your result here's my rough draft.

expr1 = 225 a^4 + 600 a^3 b^2 + 520 a^2 b^4 + 160 a b^6 + 16 b^8 + 1200 a^4 b x + 
   2080 a^3 b^3 x + 960 a^2 b^5 x + 128 a b^7 x + 600 a^5 x^2 + 3120 a^4 b^2 x^2 + 
   2400 a^3 b^4 x^2 + 448 a^2 b^6 x^2 + 2080 a^5 b x^3 + 3200 a^4 b^3 x^3 + 
   896 a^3 b^5 x^3 + 520 a^6 x^4 + 2400 a^5 b^2 x^4 + 1120 a^4 b^4 x^4 + 960 a^6 b x^5 + 
   896 a^5 b^3 x^5 + 160 a^7 x^6 + 448 a^6 b^2 x^6 + 128 a^7 b x^7 + 16 a^8 x^8;

id /. ToRules @ Reduce[{expr1 == id, zz == (b + a x)}, id, b] // Quiet

% /. zz -> (b + a x)

(15 a^2 + 20 a zz^2 + 4 zz^4)^2

(15 a^2 + 20 a (b + a x)^2 + 4 (b + a x)^4)^2

id and zz are just arbitrary placeholders for the transformation.

Applying the same process to your second expression gives:

108056025 a^10 + 1364774400 a^16 x^12 - 1364774400 a^18 x^14 - 
 1364774400 a^15 x^11 (b + a x) + 4094323200 a^17 x^13 (b + a x) + 
 720373500 a^9 (b + a x)^2 - 4094323200 a^16 x^12 (b + a x)^2 + 
 1364774400 a^15 x^11 (b + a x)^3 + 1776921300 a^8 (b + a x)^4 + 
 2085652800 a^7 (b + a x)^6 + 1335549600 a^6 (b + a x)^8 + 500734080 a^5 (b + a x)^10 + 
 113731200 a^4 (b + a x)^12 + 15713280 a^3 (b + a x)^14 + 1281280 a^2 (b + a x)^16 + 
 56320 a (b + a x)^18 + 1024 (b + a x)^20