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How do you simplify #(t^3+8)/(3t^2+t-10)#?

Answer 1

#color(blue)(1/3t-1/9+[color(white)(.)(31t+92)/(27t^2+9t-90) color(white)(.)] #

Write #t^3+8" as "t^3+0t^2+0t+8# for simplicity of comparison

The #0t^2" and "0t# have no value. They are just 'place holders' to make formatting of this solution easier.

#" "t^3+0t^2+0t+8# #color(red)(1/3t)(3t^2+t-10) -> " "ul(t^3+1/3t^2-10/3t)" "larr" Subtract"# #" "0-1/3t^2+10/3t+8 # #color(red)(-1/9)(3t^2+t-10) ->" " ul(-1/3t^2-1/9t+10/9 ) " "larr" Subtract"# #color(red)(" Remainder" -> 0+31/9t+92/9)# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(red)(1/3t-1/9+[color(white)(.)(31t+92)/9 -:(3t^2+t-10)color(white)(.)] #

#color(blue)(1/3t-1/9+[color(white)(.)(31t+92)/(27t^2+9t-90) color(white)(.)] #

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Answer 2

Jameson Allen

To simplify the expression (t^3+8)/(3t^2+t-10), you can factor the numerator and denominator. The numerator can be factored as the sum of cubes: (t+2)(t^2-2t+4). The denominator can be factored as (3t-5)(t+2). Cancel out the common factor of (t+2) from the numerator and denominator. The simplified expression is (t^2-2t+4)/(3t-5).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.