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Mathematics 16 Online
OpenStudy (anonymous):

expand using the binomial theorem: (x^2-x/2)^5

OpenStudy (loser66):

what is the binomial theorem?

OpenStudy (anonymous):

here is the expansion x^10-(5 x^9)/2+(5 x^8)/2-(5 x^7)/4+(5 x^6)/16-x^5/32 @tymckata

OpenStudy (anonymous):

wait here is the full expansion

OpenStudy (anonymous):

it is long so pls read

OpenStudy (anonymous):

= (x^2-x/2)^5 = (x^2-x/2)^5 = (x^2-x/2) (x^2-x/2)^4 = x^2 (x^2-x/2)^4-1/2 x (x^2-x/2)^4 = x^2 (x^2-x/2) (x^2-x/2)^3-1/2 x (x^2-x/2) (x^2-x/2)^3 = x^2 (x^2 (x^2-x/2)^3-1/2 x (x^2-x/2)^3)-1/2 x (x^2 (x^2-x/2)^3-1/2 x (x^2-x/2)^3) = x^2 x^2 (x^2-x/2)^3+1/2 x^2 (-1) x (x^2-x/2)^3-1/2 (x x^2 (x^2-x/2)^3+1/2 x (-1) x (x^2-x/2)^3) = x^2 x^2 (x^2-x/2)^3-1/2 x x^2 (x^2-x/2)^3-1/2 (x x^2 (x^2-x/2)^3-1/2 x x (x^2-x/2)^3) = x^(2+2) (x^2-x/2)^3-1/2 x^(1+2) (x^2-x/2)^3-1/2 (x^(1+2) (x^2-x/2)^3-1/2 x^(1+1) (x^2-x/2)^3) = x^4 (x^2-x/2)^3-1/2 x^3 (x^2-x/2)^3-1/2 (x^3 (x^2-x/2)^3-1/2 x^2 (x^2-x/2)^3) = x^4 (x^2-x/2) (x^2-x/2)^2-1/2 x^3 (x^2-x/2) (x^2-x/2)^2-1/2 (x^3 (x^2-x/2) (x^2-x/2)^2-1/2 x^2 (x^2-x/2) (x^2-x/2)^2) = x^4 (x^2 (x^2-x/2)^2-1/2 x (x^2-x/2)^2)-1/2 x^3 (x^2 (x^2-x/2)^2-1/2 x (x^2-x/2)^2)-1/2 (x^3 (x^2 (x^2-x/2)^2-1/2 x (x^2-x/2)^2)-1/2 x^2 (x^2 (x^2-x/2)^2-1/2 x (x^2-x/2)^2)) = x^4 x^2 (x^2-x/2)^2+1/2 x^4 (-1) x (x^2-x/2)^2-1/2 (x^3 x^2 (x^2-x/2)^2+1/2 x^3 (-1) x (x^2-x/2)^2)-1/2 (x^3 x^2 (x^2-x/2)^2+1/2 x^3 (-1) x (x^2-x/2)^2-1/2 (x^2 x^2 (x^2-x/2)^2+1/2 x^2 (-1) x (x^2-x/2)^2)) = x^2 x^4 (x^2-x/2)^2-1/2 x x^4 (x^2-x/2)^2-1/2 (x^2 x^3 (x^2-x/2)^2-1/2 x x^3 (x^2-x/2)^2)-1/2 (x^2 x^3 (x^2-x/2)^2-1/2 x x^3 (x^2-x/2)^2-1/2 (x^2 x^2 (x^2-x/2)^2-1/2 x x^2 (x^2-x/2)^2)) = x^(2+4) (x^2-x/2)^2-1/2 x^(1+4) (x^2-x/2)^2-1/2 (x^(2+3) (x^2-x/2)^2-1/2 x^(1+3) (x^2-x/2)^2)-1/2 (x^(2+3) (x^2-x/2)^2-1/2 x^(1+3) (x^2-x/2)^2-1/2 (x^(2+2) (x^2-x/2)^2-1/2 x^(1+2) (x^2-x/2)^2)) = x^6 (x^2-x/2)^2-1/2 x^5 (x^2-x/2)^2-1/2 (x^5 (x^2-x/2)^2-1/2 x^4 (x^2-x/2)^2)-1/2 (x^5 (x^2-x/2)^2-1/2 x^4 (x^2-x/2)^2-1/2 (x^4 (x^2-x/2)^2-1/2 x^3 (x^2-x/2)^2)) = x^6 (x^2-x/2) (x^2-x/2)-1/2 x^5 (x^2-x/2) (x^2-x/2)-1/2 (x^5 (x^2-x/2) (x^2-x/2)-1/2 x^4 (x^2-x/2) (x^2-x/2))-1/2 (x^5 (x^2-x/2) (x^2-x/2)-1/2 x^4 (x^2-x/2) (x^2-x/2)-1/2 (x^4 (x^2-x/2) (x^2-x/2)-1/2 x^3 (x^2-x/2) (x^2-x/2))) = x^6 (x^2 (x^2-x/2)-1/2 x (x^2-x/2))-1/2 x^5 (x^2 (x^2-x/2)-1/2 x (x^2-x/2))-1/2 (x^5 (x^2 (x^2-x/2)-1/2 x (x^2-x/2))-1/2 x^4 (x^2 (x^2-x/2)-1/2 x (x^2-x/2)))-1/2 (x^5 (x^2 (x^2-x/2)-1/2 x (x^2-x/2))-1/2 x^4 (x^2 (x^2-x/2)-1/2 x (x^2-x/2))-1/2 (x^4 (x^2 (x^2-x/2)-1/2 x (x^2-x/2))-1/2 x^3 (x^2 (x^2-x/2)-1/2 x (x^2-x/2)))) = x^6 x^2 (x^2-x/2)+1/2 x^6 (-1) x (x^2-x/2)-1/2 (x^5 x^2 (x^2-x/2)+1/2 x^5 (-1) x (x^2-x/2))-1/2 (x^5 x^2 (x^2-x/2)+1/2 x^5 (-1) x (x^2-x/2)-1/2 (x^4 x^2 (x^2-x/2)+1/2 x^4 (-1) x (x^2-x/2)))-1/2 (x^5 x^2 (x^2-x/2)+1/2 x^5 (-1) x (x^2-x/2)-1/2 (x^4 x^2 (x^2-x/2)+1/2 x^4 (-1) x (x^2-x/2))-1/2 (x^4 x^2 (x^2-x/2)+1/2 x^4 (-1) x (x^2-x/2)-1/2 (x^3 x^2 (x^2-x/2)+1/2 x^3 (-1) x (x^2-x/2)))) = x^2 x^6 (x^2-x/2)-1/2 x x^6 (x^2-x/2)-1/2 (x^2 x^5 (x^2-x/2)-1/2 x x^5 (x^2-x/2))-1/2 (x^2 x^5 (x^2-x/2)-1/2 x x^5 (x^2-x/2)-1/2 (x^2 x^4 (x^2-x/2)-1/2 x x^4 (x^2-x/2)))-1/2 (x^2 x^5 (x^2-x/2)-1/2 x x^5 (x^2-x/2)-1/2 (x^2 x^4 (x^2-x/2)-1/2 x x^4 (x^2-x/2))-1/2 (x^2 x^4 (x^2-x/2)-1/2 x x^4 (x^2-x/2)-1/2 (x^2 x^3 (x^2-x/2)-1/2 x x^3 (x^2-x/2)))) = x^(2+6) (x^2-x/2)-1/2 x^(1+6) (x^2-x/2)-1/2 (x^(2+5) (x^2-x/2)-1/2 x^(1+5) (x^2-x/2))-1/2 (x^(2+5) (x^2-x/2)-1/2 x^(1+5) (x^2-x/2)-1/2 (x^(2+4) (x^2-x/2)-1/2 x^(1+4) (x^2-x/2)))-1/2 (x^(2+5) (x^2-x/2)-1/2 x^(1+5) (x^2-x/2)-1/2 (x^(2+4) (x^2-x/2)-1/2 x^(1+4) (x^2-x/2))-1/2 (x^(2+4) (x^2-x/2)-1/2 x^(1+4) (x^2-x/2)-1/2 (x^(2+3) (x^2-x/2)-1/2 x^(1+3) (x^2-x/2)))) = x^8 (x^2-x/2)-1/2 x^7 (x^2-x/2)-1/2 (x^7 (x^2-x/2)-1/2 x^6 (x^2-x/2))-1/2 (x^7 (x^2-x/2)-1/2 x^6 (x^2-x/2)-1/2 (x^6 (x^2-x/2)-1/2 x^5 (x^2-x/2)))-1/2 (x^7 (x^2-x/2)-1/2 x^6 (x^2-x/2)-1/2 (x^6 (x^2-x/2)-1/2 x^5 (x^2-x/2))-1/2 (x^6 (x^2-x/2)-1/2 x^5 (x^2-x/2)-1/2 (x^5 (x^2-x/2)-1/2 x^4 (x^2-x/2)))) = x^8 x^2+1/2 x^8 (-1) x-1/2 (x^7 x^2+1/2 x^7 (-1) x)-1/2 (x^7 x^2+1/2 x^7 (-1) x-1/2 (x^6 x^2+1/2 x^6 (-1) x))-1/2 (x^7 x^2+1/2 x^7 (-1) x-1/2 (x^6 x^2+1/2 x^6 (-1) x)-1/2 (x^6 x^2+1/2 x^6 (-1) x-1/2 (x^5 x^2+1/2 x^5 (-1) x)))-1/2 (x^7 x^2+1/2 x^7 (-1) x-1/2 (x^6 x^2+1/2 x^6 (-1) x)-1/2 (x^6 x^2+1/2 x^6 (-1) x-1/2 (x^5 x^2+1/2 x^5 (-1) x))-1/2 (x^6 x^2+1/2 x^6 (-1) x-1/2 (x^5 x^2+1/2 x^5 (-1) x)-1/2 (x^5 x^2+1/2 x^5 (-1) x-1/2 (x^4 x^2+1/2 x^4 (-1) x)))) = x^2 x^8-(x x^8)/2-1/2 (x^2 x^7-(x x^7)/2)-1/2 (x^2 x^7-(x x^7)/2-1/2 (x^2 x^6-(x x^6)/2))-1/2 (x^2 x^7-(x x^7)/2-1/2 (x^2 x^6-(x x^6)/2)-1/2 (x^2 x^6-(x x^6)/2-1/2 (x^2 x^5-(x x^5)/2)))-1/2 (x^2 x^7-(x x^7)/2-1/2 (x^2 x^6-(x x^6)/2)-1/2 (x^2 x^6-(x x^6)/2-1/2 (x^2 x^5-(x x^5)/2))-1/2 (x^2 x^6-(x x^6)/2-1/2 (x^2 x^5-(x x^5)/2)-1/2 (x^2 x^5-(x x^5)/2-1/2 (x^2 x^4-(x x^4)/2)))) = x^(2+8)-x^(1+8)/2-1/2 (x^(2+7)-x^(1+7)/2)-1/2 (x^(2+7)-x^(1+7)/2-1/2 (x^(2+6)-x^(1+6)/2))-1/2 (x^(2+7)-x^(1+7)/2-1/2 (x^(2+6)-x^(1+6)/2)-1/2 (x^(2+6)-x^(1+6)/2-1/2 (x^(2+5)-x^(1+5)/2)))-1/2 (x^(2+7)-x^(1+7)/2-1/2 (x^(2+6)-x^(1+6)/2)-1/2 (x^(2+6)-x^(1+6)/2-1/2 (x^(2+5)-x^(1+5)/2))-1/2 (x^(2+6)-x^(1+6)/2-1/2 (x^(2+5)-x^(1+5)/2)-1/2 (x^(2+5)-x^(1+5)/2-1/2 (x^(2+4)-x^(1+4)/2)))) = x^10-x^9/2-1/2 (x^9-x^8/2)-1/2 (x^9-x^8/2-1/2 (x^8-x^7/2))-1/2 (x^9-x^8/2-1/2 (x^8-x^7/2)-1/2 (x^8-x^7/2-1/2 (x^7-x^6/2)))-1/2 (x^9-x^8/2-1/2 (x^8-x^7/2)-1/2 (x^8-x^7/2-1/2 (x^7-x^6/2))-1/2 (x^8-x^7/2-1/2 (x^7-x^6/2)-1/2 (x^7-x^6/2-1/2 (x^6-x^5/2)))) = x^10-x^9/2-x^9/2--x^8/(2 2)-x^9/2--x^8/(2 2)--(x^8-x^7/2)/(2 2)-x^9/2--x^8/(2 2)--(x^8-x^7/2)/(2 2)--(x^8-x^7/2-1/2 (x^7-x^6/2))/(2 2)-x^9/2--x^8/(2 2)--(x^8-x^7/2)/(2 2)--(x^8-x^7/2-1/2 (x^7-x^6/2))/(2 2)--(x^8-x^7/2-1/2 (x^7-x^6/2)-1/2 (x^7-x^6/2-1/2 (x^6-x^5/2)))/(2 2) = x^10-x^9/2-x^9/2+(-1/2)^(1+1) x^8-x^9/2+(-1/2)^(1+1) x^8+(-1/2)^(1+1) (x^8-x^7/2)-x^9/2+(-1/2)^(1+1) x^8+(-1/2)^(1+1) (x^8-x^7/2)+(-1/2)^(1+1) (x^8-x^7/2-1/2 (x^7-x^6/2))-x^9/2+(-1/2)^(1+1) x^8+(-1/2)^(1+1) (x^8-x^7/2)+(-1/2)^(1+1) (x^8-x^7/2-1/2 (x^7-x^6/2))+(-1/2)^(1+1) (x^8-x^7/2-1/2 (x^7-x^6/2)-1/2 (x^7-x^6/2-1/2 (x^6-x^5/2))) = x^10+5/2 (-1) x^9+(4 x^8)/4+3/4 (x^8-x^7/2)+2/4 (x^8-x^7/2-1/2 (x^7-x^6/2))+1/4 (x^8-x^7/2-1/2 (x^7-x^6/2)-1/2 (x^7-x^6/2-1/2 (x^6-x^5/2))) = x^10-(5 x^9)/2+(4 x^8)/4+3/4 (x^8-x^7/2)+2/4 (x^8-x^7/2-1/2 (x^7-x^6/2))+1/4 (x^8-x^7/2-1/2 (x^7-x^6/2)-1/2 (x^7-x^6/2-1/2 (x^6-x^5/2))) = x^10-(5 x^9)/2+x^8+3/4 (x^8-x^7/2)+1/2 (x^8-x^7/2-1/2 (x^7-x^6/2))+1/4 (x^8-x^7/2-1/2 (x^7-x^6/2)-1/2 (x^7-x^6/2-1/2 (x^6-x^5/2))) = x^10-(5 x^9)/2+x^8+(3 x^8)/4+(3 (-1) x^7)/(4 2)+x^8/2-x^7/(2 2)-(x^7-x^6/2)/(2 2)+x^8/4-x^7/(4 2)-(x^7-x^6/2)/(4 2)-(x^7-x^6/2-1/2 (x^6-x^5/2))/(4 2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/(2 4)+x^8/2-x^7/(2 2)-(x^7-x^6/2)/(2 2)+x^8/4-x^7/(2 4)-(x^7-x^6/2)/(2 4)-(x^7-x^6/2-1/2 (x^6-x^5/2))/(2 4) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/8+x^8/2-x^7/4-1/4 (x^7-x^6/2)+x^8/4-x^7/8-1/8 (x^7-x^6/2)-1/8 (x^7-x^6/2-1/2 (x^6-x^5/2)) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/8+x^8/2-x^7/4-x^7/4--x^6/(4 2)+x^8/4-x^7/8-x^7/8--x^6/(8 2)-x^7/8--x^6/(8 2)--(x^6-x^5/2)/(8 2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/8+x^8/2-x^7/4-x^7/4--x^6/(2 4)+x^8/4-x^7/8-x^7/8--x^6/(2 8)-x^7/8--x^6/(2 8)--(x^6-x^5/2)/(2 8) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/8+x^8/2-x^7/4-x^7/4+x^6/8+x^8/4-x^7/8-x^7/8+x^6/16-x^7/8+x^6/16+1/16 (x^6-x^5/2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/8+x^8/2+2/4 (-1) x^7+x^6/8+x^8/4+3/8 (-1) x^7+(2 x^6)/16+1/16 (x^6-x^5/2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/8+x^8/2-(2 x^7)/4+x^6/8+x^8/4-(3 x^7)/8+(2 x^6)/16+1/16 (x^6-x^5/2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/8+x^8/2-x^7/2+x^6/8+x^8/4-(3 x^7)/8+x^6/8+1/16 (x^6-x^5/2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4+2/8 (-3) x^7+x^8/2-x^7/2+(2 x^6)/8+x^8/4+1/16 (x^6-x^5/2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 2 x^7)/8+x^8/2-x^7/2+(2 x^6)/8+x^8/4+1/16 (x^6-x^5/2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/4+x^8/2-x^7/2+x^6/4+x^8/4+1/16 (x^6-x^5/2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/4+x^8/2-x^7/2+x^6/4+x^8/4+x^6/16-x^5/(16 2) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/4+x^8/2-x^7/2+x^6/4+x^8/4+x^6/16-x^5/(2 16) = x^10-(5 x^9)/2+x^8+(3 x^8)/4-(3 x^7)/4+x^8/2-x^7/2+x^6/4+x^8/4+x^6/16-x^5/32 Answer: | | = -x^5/32+(5 x^6)/16-(5 x^7)/4+(5 x^8)/2-(5 x^9)/2+x^10

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