Maths-
General
Easy

Question

Write an equivalent expression , state the domain:
fraction numerator 3 straight x squared plus 15 straight x over denominator straight x squared plus 3 straight x minus 10 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 is. 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 find an equivalent expression and domain for the given expression.

The correct answer is: Hence, the domain is, (-∞,2)∪(2,∞).


    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 3 x squared plus 15 x over denominator x squared plus 3 x minus 10 end fraction equals fraction numerator 3 x left parenthesis x plus 5 right parenthesis over denominator x squared plus 5 x minus 2 x minus 10 end fraction end cell row cell equals fraction numerator 3 x left parenthesis x plus 5 right parenthesis over denominator x left parenthesis x plus 5 right parenthesis minus 2 left parenthesis x plus 5 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space end cell row cell space space space equals fraction numerator 3 x left parenthesis x plus 5 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x plus 5 right parenthesis end fraction space space space space space space space space space space space space space space 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 3 x over denominator x minus 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end cell end table
    Hence, the equivalent expression of fraction numerator 3 straight x squared plus 15 straight x over denominator straight x squared plus 3 straight x minus 10 end fraction text  is  end text fraction numerator 3 straight x over denominator straight x minus 2 end fraction.
    Step 2 of 2:
    When we find the domain, we have to exclude the values for which the denominator attains a zero value. So, we have:
    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 x minus 2 equals 0 end cell row cell x equals 2 space space space space space space end cell end table
    That is,  x = 2 must be excluded.
    Hence, the domain is, left parenthesis negative straight infinity comma 2 right parenthesis union left parenthesis 2 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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