Maths-
General
Easy

Question

What is the simplified form of each rational expression ? What is the domain ?
fraction numerator straight x cubed plus 9 straight x squared minus 10 straight x over denominator straight x cubed minus 9 straight x squared minus 10 straight x end fraction

hintHint:

The expansion of left parenthesis straight x plus straight a right parenthesis left parenthesis straight x plus straight b right parenthesis text  is  end text straight x squared plus left parenthesis straight a plus straight b right parenthesis straight x plus ab. We have to state the domain of a rational expression while simplifying them because we must exclude zeros of a denominator as dividing with zero is not defined.
We are asked to simplify the expression and give its domain.

The correct answer is: Thus, the domain is, (-∞,-1) ∪ (-1,10) ∪ (10,∞)


    Step 1 of 2:
    Simplify the expression and cancel out the common factors,
    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell fraction numerator x cubed plus 9 x squared minus 10 x over denominator x cubed minus 9 x squared minus 10 x end fraction equals fraction numerator x open parentheses x squared plus 9 x minus 10 close parentheses over denominator x open parentheses x squared minus 9 x minus 10 close parentheses end fraction end cell row cell equals fraction numerator x open parentheses x squared plus 10 x minus x minus 10 close parentheses over denominator x open parentheses x squared minus 10 x plus x minus 10 close parentheses end fraction space space space space space space space space space space space space space space space space space space end cell row cell equals fraction numerator x left parenthesis x left parenthesis x plus 10 right parenthesis minus 1 left parenthesis x plus 10 right parenthesis right parenthesis over denominator x left parenthesis x left parenthesis x minus 10 right parenthesis plus 1 left parenthesis x minus 10 right parenthesis right parenthesis end fraction space space space space space space space space space space space space space end cell end table
    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell equals fraction numerator x left parenthesis x minus 1 right parenthesis left parenthesis x plus 10 right parenthesis over denominator x left parenthesis x plus 1 right parenthesis left parenthesis x minus 10 right parenthesis end fraction end cell row cell equals fraction numerator left parenthesis x minus 1 right parenthesis left parenthesis x plus 10 right parenthesis over denominator left parenthesis x plus 1 right parenthesis left parenthesis x minus 10 right parenthesis end fraction space space space end cell end table
    Thus, the simplified expression is fraction numerator left parenthesis x minus 1 right parenthesis left parenthesis x plus 10 right parenthesis over denominator left parenthesis x plus 1 right parenthesis left parenthesis x minus 10 right parenthesis end fraction
    Step 2 of 2:
    Domain of a rational expression should exclude the values of which the denominator gets a zero value.
    Hence, we have:
    table attributes columnalign right columnspacing 0em end attributes row cell left parenthesis straight x plus 1 right parenthesis left parenthesis straight x minus 10 right parenthesis equals 0 end cell row cell straight x equals negative 1 space space space space space space space space space space space space space space space space space end cell row cell straight x equals 10 space space space space space space space space space space space space space space space space space space end cell end table
    Thus, the domain is,left parenthesis negative straight infinity comma negative 1 right parenthesis union left parenthesis negative 1 comma 10 right parenthesis union left parenthesis 10 comma straight infinity right parenthesis.

    We have to state the domain of a rational expression while simplifying them because we must exclude zeros of a denominator as dividing with zero is not defined.

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