Maths-
General
Easy

Question

Find the product and the domain ?
fraction numerator 2 straight x squared minus 10 straight x over denominator left parenthesis straight x minus 5 right parenthesis open parentheses straight x squared minus 1 close parentheses end fraction cross times 3 straight x squared plus 4 straight x plus 1

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 space end cell end table
We are asked to find the product and the domain of the given expression.

The correct answer is: Thus, the domain of the expression is, (-∞,1)∪(1,∞).


    Step 1 of 2:
    Simplify the expression and cancel out common terms;
    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 2 x squared minus 10 x over denominator left parenthesis x minus 5 right parenthesis open parentheses x squared minus 1 close parentheses end fraction cross times space 3 x squared plus 4 x plus 1 equals fraction numerator 2 x left parenthesis x minus 5 right parenthesis over denominator left parenthesis x minus 5 right parenthesis open parentheses x squared minus 1 squared close parentheses end fraction cross times 3 x squared plus 3 x plus x plus 1 end cell row cell equals fraction numerator 2 x left parenthesis x minus 5 right parenthesis over denominator left parenthesis x minus 5 right parenthesis left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction cross times 3 x left parenthesis x plus 1 right parenthesis plus 1 left parenthesis x plus 1 right parenthesis 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 equals fraction numerator 2 x left parenthesis x minus 5 right parenthesis over denominator left parenthesis x minus 5 right parenthesis left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction cross times left parenthesis 3 x plus 1 right parenthesis left parenthesis x plus 1 right parenthesis space space space space space space space space end cell row cell equals fraction numerator 2 x left parenthesis 3 x plus 1 right parenthesis over denominator left parenthesis x minus 1 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 end cell end table
    Thus, the simplified expression is fraction numerator 2 straight x left parenthesis 3 straight x plus 1 right parenthesis over denominator left parenthesis straight x minus 1 right parenthesis end fraction
    Step 2 of 2:
    The domain of a rational expression should exclude the values for which the denominator attains a zero values. So, when
    table attributes columnalign right columnspacing 0em end attributes row cell straight x minus 1 equals 0 space space space space space space space space space space space space end cell row cell straight x equals 1 space space space space space space space space space space space space space space space space space end cell end table space space space space space space space space space space space space
    Thus, the domain of the expression is, (-∞,1)∪(1,∞).

    Division by a zero value is not defined in mathematics. Hence, we cannot divide a number by zero.

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