Doo 2 3 3 X 2

• 2021 Sea-Doo Spark Trixx 3-up IBR Jet Ski Trades Welcome - $8,599 (VE Motoplex Sioux Falls SD)

  Phone: 605-221-40002021 Sea-Doo Spark Trixx 3-up IBR Jet Ski Trades Welcome Use the following link to view more information about this vehicle: https://www.verneidemotoplex.com/lc/9270902JUST IN 2021 TRIXX 3-UP IBR Green Includes Cover, Anchor, Fire Extinguisher registrat ..

• 2021 SEA-DOO SPARK TRIXX 2 UP!!! - $7,699 (Osage Beach)

  2021 SEA-DOO SPARK TRIXX 2UP!!!ONLY $7699WE ARE LOCATED RIGHT OFF PASSOVER ON FORMULA DRAND BY WATER AT THE 19 MILE MARKER IN FORMULA's COVE JUST SOUTH OF THE GRAVOIS ARM..BUY ONE TODAY AND LET US LAUNCH YOU IN THE WATER ON SITE..YACHT CLUB POWERSPORTS ..

• 2021 SEA-DOO SPARK TRIXX 2 UP WITH AUDIO!!!!! - $8,299 (Osage Beach)

  2021 SEA-DOO SPARK TRIXX 2 PASSENGER WITH AUDIO!!!ONLY $8299 WE ARE LOCATED RIGHT OFF PASSOVER ON FORMULA DRAND BY WATER AT THE 19 MILE MARKER IN FORMULA's COVE JUST SOUTH OF THE GRAVOIS ARM..BUY ONE TODAY AND LET US LAUNCH YOU IN THE WATER ON SITE..YACHT ..

• 2021 Sea Doo Spark Trixx 3-UP iBR w/ Audio Personal Watercraft - $8,900 (Olsen's Outdoor Power-Carroll)

  Olsen's Outdoor Power - CarrollAsk for: Sales☎ (712) 775-400019072 Kittyhawk Ave Carroll, IA 51401Description:Model 66MFCompare against Jet Ski & WaveRunner.Olsen's Outdoor Power Carroll, serving Carroll, Ames, Ft. Dodge, Storm Lake, Deniso ..

• 2021 Sea Doo Spark Trixx 3-UP iBR Personal Watercraft - $8,300 (Olsen's Outdoor Power-Carroll)

  Olsen's Outdoor Power - CarrollAsk for: Sales☎ (712) 775-400019072 Kittyhawk Ave Carroll, IA 51401Description:Model 66MCCompare against Jet Ski & WaveRunner.Olsen's Outdoor Power Carroll, serving Carroll, Ames, Ft. Dodge, Storm Lake, Deniso ..

• 2021 SEA-DOO SPARK TRIXX 3 UP WITH AUDIO!!!!! - $8,899 (Osage Beach)

  2021 SEA-DOO SPARK 3UP WITH AUDIO!!ONLY $8899!!WE ARE LOCATED RIGHT OFF PASSOVER ON FORMULA DRAND BY WATER AT THE 19 MILE MARKER IN FORMULA's COVE JUST SOUTH OF THE GRAVOIS ARM..BUY ONE TODAY AND LET US LAUNCH YOU IN THE WATER ON SITE..YACHT CLUB POWERSPOR ..

• 2019 Sea-Doo Spark Trixx 2-up Rotax 900 H O ACE IBR Sound Syste - $8049 (BAD CREDIT OK)

  2019 Sea-Doo Spark Trixx 2-up Rotax 900 H O ACE IBR Sound Syste - (ORANGE CRUSH/ CHILI PEPPER) STOCK #: DWK77631B Available at: Broward Motorsports Treasure Coast ☎ Phone: (772) 208-6051 Year: 2019 Make: Sea-Doo Model: Spark Trixx 2 ..

• 2019 Sea-Doo Spark Trixx 2-up Rotax 900 H O ACE IBR - $7708 (EZ FINANCING)

  2019 Sea-Doo Spark Trixx 2-up Rotax 900 H O ACE IBR - () STOCK #: YDV71575L819 Available at: Water World Boat & Powersport ☎ Phone: (541) 826-5535 Year: 2019 Make: Sea-Doo Model: Spark Trixx 2-up Rotax 900 H O ACE IBR Col ..

• Doo 2 3 3 X 2
• X 3 X 2 2 X 7
• Doo 2 3 3 X 2 0cm
• X X 3 1 2

1. 3DOo / Diamond 1 79LP / 82W 75L Win Ratio 52% / Thresh - 30W 39L Win Ratio 43%, Senna - 27W 20L Win Ratio 57%, Twisted Fate - 20W 16L Win Ratio 56%, Kai'Sa - 17W 19L Win Ratio 47%, Shen - 21W 10L Win Ratio 68%.
2. With Frank Welker, Grey Griffin, Kate Micucci, Matthew Lillard. The Scooby gang continues to solve mysteries during what could be their last summer together, but monsters and ghouls get in the way of completing their adventures.

Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history.

How do you factor # x^4 + x^3 + x^2 + x + 1#?

1 Answer

#x^4+x^3+x^2+x+1#

#=(x^2+(1/2+sqrt(5)/2)x+1)(x^2+(1/2-sqrt(5)/2)x+1)#

Explanation:

This quartic has four zeros, which are the non-Real Complex #5#th roots of #1#, as we can see from:

#(x-1)(x^4+x^3+x^2+x+1) = x^5-1#

So if we wanted to factor this polynomial as a product of linear factors with Complex coefficients then we could write:

#x^4+x^3+x^2+x+1#

#=(x-(cos((2pi)/5) + i sin((2pi)/5))) * (x-(cos((4pi)/5) + i sin((4pi)/5))) * (x-(cos((6pi)/5) + i sin((6pi)/5))) * (x-(cos((8pi)/5) + i sin((8pi)/5)))#

A cleaner algebraic approach is to notice that due to the symmetry of the coefficients, if #x=r# is a zero of #x^4+x^3+x^2+x+1#, then #x=1/r# is also a zero.

Hence there is a factorisation in the form:

Doo 2 3 3 X 2

#x^4+x^3+x^2+x+1#

#=(x-r_1)(x-1/r_1)(x-r_2)(x-1/r_2)#

#=(x^2-(r_1+1/r_1)x+1)(x^2-(r_2+1/r_2)x+1)#

So let's look for a factorisation:

#x^4+x^3+x^2+x+1#

#=(x^2+ax+1)(x^2+bx+1)#

#=x^4+(a+b)x^3+(2+ab)x^2+(a+b)x+1#

X 3 X 2 2 X 7

Equating coefficients we find:

#a+b = 1#

#2+ab=1#, so #ab = -1# and #b=-1/a#

Substituting #b=-1/a# in #a+b=1# we get:

#a-1/a = 1#

Hence:

#a^2-a-1 = 0#

Doo 2 3 3 X 2 0cm

Using the quadratic formula, we can deduce:

#a = 1/2 +- sqrt(5)/2#

Since our derivation was symmetric in #a# and #b#, one of these roots can be used for #a# and the other for #b#, to find:

X X 3 1 2

#x^4+x^3+x^2+x+1#

#=(x^2+(1/2+sqrt(5)/2)x+1)(x^2+(1/2-sqrt(5)/2)x+1)# Postico 1 3 2 – a modern postgresql client create.

If we want to factor further, use the quadratic formula on each of these quadratic factors to find the linear factors with Complex coefficients.

Related questions