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Popular Algebra >

expand (2-x)^{19}

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Solution

expand (2−x)19

Solution

524288−4980736x+22413312x2−63504384x3+127008768x4−190513152x5+222265344x6−206389248x7+154791936x8−94595072x9+47297536x10−19348992x11+6449664x12−1736448x13+372096x14−62016x15+7752x16−684x17+38x18−x19
Solution steps
(2−x)19
Apply binomial theorem: (a+b)n=i=0∑n​(in​)a(n−i)bia=2,b=−x
=i=0∑19​(i19​)⋅2(19−i)(−x)i
Expand summation
=0!(19−0)!19!​⋅219(−x)0+1!(19−1)!19!​⋅218(−x)1+2!(19−2)!19!​⋅217(−x)2+3!(19−3)!19!​⋅216(−x)3+4!(19−4)!19!​⋅215(−x)4+5!(19−5)!19!​⋅214(−x)5+6!(19−6)!19!​⋅213(−x)6+7!(19−7)!19!​⋅212(−x)7+8!(19−8)!19!​⋅211(−x)8+9!(19−9)!19!​⋅210(−x)9+10!(19−10)!19!​⋅29(−x)10+11!(19−11)!19!​⋅28(−x)11+12!(19−12)!19!​⋅27(−x)12+13!(19−13)!19!​⋅26(−x)13+14!(19−14)!19!​⋅25(−x)14+15!(19−15)!19!​⋅24(−x)15+16!(19−16)!19!​⋅23(−x)16+17!(19−17)!19!​⋅22(−x)17+18!(19−18)!19!​⋅21(−x)18+19!(19−19)!19!​⋅20(−x)19
Simplify 0!(19−0)!19!​⋅219(−x)0:524288
Simplify 1!(19−1)!19!​⋅218(−x)1:−4980736x
Simplify 2!(19−2)!19!​⋅217(−x)2:22413312x2
Simplify 3!(19−3)!19!​⋅216(−x)3:−63504384x3
Simplify 4!(19−4)!19!​⋅215(−x)4:127008768x4
Simplify 5!(19−5)!19!​⋅214(−x)5:−190513152x5
Simplify 6!(19−6)!19!​⋅213(−x)6:222265344x6
Simplify 7!(19−7)!19!​⋅212(−x)7:−206389248x7
Simplify 8!(19−8)!19!​⋅211(−x)8:154791936x8
Simplify 9!(19−9)!19!​⋅210(−x)9:−94595072x9
Simplify 10!(19−10)!19!​⋅29(−x)10:47297536x10
Simplify 11!(19−11)!19!​⋅28(−x)11:−19348992x11
Simplify 12!(19−12)!19!​⋅27(−x)12:6449664x12
Simplify 13!(19−13)!19!​⋅26(−x)13:−1736448x13
Simplify 14!(19−14)!19!​⋅25(−x)14:372096x14
Simplify 15!(19−15)!19!​⋅24(−x)15:−62016x15
Simplify 16!(19−16)!19!​⋅23(−x)16:7752x16
Simplify 17!(19−17)!19!​⋅22(−x)17:−684x17
Simplify 18!(19−18)!19!​⋅21(−x)18:38x18
Simplify 19!(19−19)!19!​⋅20(−x)19:−x19
=524288−4980736x+22413312x2−63504384x3+127008768x4−190513152x5+222265344x6−206389248x7+154791936x8−94595072x9+47297536x10−19348992x11+6449664x12−1736448x13+372096x14−62016x15+7752x16−684x17+38x18−x19

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