Factor #t^(1/2) + 6t^(-1/2) -t^(-3/2) - 6t^(-5/2)# completely so that factors of two or more terms contain no negative exponents or fractional coefficients?

1 Answer
Jim G.
Feb 9, 2018

#t^(-5/2)(t-1)(t+1)(t+6)#

Explanation:

#"take out a "color(blue)"common factor of "t^(-5/2)#

#rArrt^(-5/2)(t^3+6t^2-t-6)#

#color(red)"factorise "t^3+6t^2-t-6#

#"note the coefficients "1+6-1-6=0#

#rArr(t-1)" is a factor"#

#"divide "t^3+6t^2-t-6" by "(t-1)#

#color(red)(t^2)(t-1)color(magenta)(+t^2)+6t^2-t-6#

#=color(red)(t^2)(t-1)color(red)(+7t)(t-1)color(magenta)(+7t)-t-6#

#=color(red)(t^2)(t-1)color(red)(+7t)(t-1)color(red)(+6)(t-1)cancel(color(magenta)(+6))cancel(-6)#

#rArr(t^3+6t^2-t-6)/(t-1)=t^2+7t+6#

#rArrt^(1/2)+6t^(-1/2)-t^(-3/2)-6t^(-5/2)#

#=t^(-5/2)(t-1)(t^2+7t+6)#

#=t^(-5/2)(t-1)(t+1)(t+6)#

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