Maths-
General
Easy

Question

Find the Quotient and the domain .
fraction numerator straight x to the power of 4 plus straight x cubed minus 30 straight x squared over denominator straight x squared minus 3 straight x minus 18 end fraction divided by fraction numerator straight x cubed plus straight x squared minus 30 straight x over denominator straight x squared minus 36 end fraction

hintHint:

The expansions of certain identities are:
table attributes columnalign right columnspacing 0em end attributes row cell left parenthesis straight x plus straight a right parenthesis left parenthesis straight x plus straight b right parenthesis equals straight x squared plus left parenthesis straight a plus straight b right parenthesis straight x plus ab end cell row cell open parentheses straight x squared minus straight a squared close parentheses equals left parenthesis straight x minus straight a right parenthesis left parenthesis straight x plus straight a right parenthesis space space space space space space space space space space space space space space end cell end table
Finding the quotient is same as division. Dividing a number and multiplying it with a reciprocal of the number have the same effect.
We are asked to find the quotient and the domain of the expression.

The correct answer is: Hence, the domain of the expression is, (-∞,-3)∪(-3,∞)



    Step 1 of 3:
    The given expression is fraction numerator straight x to the power of 4 plus straight x cubed minus 30 straight x squared over denominator straight x squared minus 3 straight x minus 18 end fraction divided by fraction numerator straight x cubed plus straight x squared minus 30 straight x over denominator straight x squared minus 36 end fraction.
    Take the reciprocal of the second expression and then multiply them;
    fraction numerator straight x to the power of 4 plus straight x cubed minus 30 straight x squared over denominator straight x squared minus 3 straight x minus 18 end fraction cross times fraction numerator straight x squared minus 36 over denominator straight x cubed plus straight x squared minus 30 straight x end fraction
    Step 2 of 3:
    Simplify the expression and cancel out the common factors;
    table attributes columnalign right columnspacing 0em end attributes row cell fraction numerator straight x to the power of 4 plus straight x cubed minus 30 straight x squared over denominator straight x squared minus 3 straight x minus 18 end fraction cross times fraction numerator straight x squared minus 36 over denominator straight x cubed plus straight x squared minus 30 straight x end fraction equals fraction numerator straight x squared open parentheses straight x squared plus straight x minus 30 close parentheses over denominator straight x squared minus 6 straight x plus 3 straight x minus 18 end fraction cross times fraction numerator open parentheses straight x squared minus 6 squared close parentheses over denominator straight x open parentheses straight x squared plus straight x minus 30 close parentheses end fraction end cell row cell equals fraction numerator straight x left parenthesis straight x minus 6 right parenthesis left parenthesis straight x plus 6 right parenthesis over denominator straight x left parenthesis straight x minus 6 right parenthesis plus 3 left parenthesis straight x minus 6 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 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 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 space space space space space equals fraction numerator x left parenthesis x minus 6 right parenthesis left parenthesis x plus 6 right parenthesis over denominator left parenthesis x plus 3 right parenthesis left parenthesis x minus 6 right parenthesis end fraction end cell row cell equals fraction numerator x left parenthesis x plus 6 right parenthesis over denominator x plus 3 end fraction space space space space space space space space space space end cell end table
    Thus, the quotient of the expression is:fraction numerator x left parenthesis x plus 6 right parenthesis over denominator x plus 3 end fraction
    Step 3 of 3:
    The domain of a rational expression should exclude the value for which the denominator attains a zero value. Thus, 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 plus 3 equals 0 end cell row cell x equals negative 3 space space space end cell end table
    Hence, the domain of the expression is, (-∞,-3)∪(-3,∞)

    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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