Maths-
General
Easy

Question

What is the simplified form of the rational expressions ? What is the domain for which the identity between the two expressions is valid ?
fraction numerator x cubed minus 5 x squared minus 24 x over denominator x cubed plus x squared minus 72 x end fraction

hintHint:

The expansion of the expressions a squared minus x squared text  and  end text x squared plus left parenthesis a plus b right parenthesis x plus a b text  are:  end text left parenthesis a minus x right parenthesis left parenthesis a plus x right parenthesis text  and  end text left parenthesis x plus a right parenthesis left parenthesis x plus b right parenthesis: respectively. The domain is the set of input values of an expression.
We are asked to simplify the expression and find its domain.

The correct answer is: Here, the denominator of the expression cannot be zero. It is zero when: x + 5 = 0 x = - 5 .


    Step 1 of 3:
    Simplify the numerator and denominator using the identities.
    Start with numerator:
    fraction numerator 4 minus x squared over denominator x squared plus 3 x minus 10 end fraction equals fraction numerator 2 squared minus x squared over denominator x squared plus 3 x minus 10 end fraction
    equals fraction numerator left parenthesis 2 minus x right parenthesis left parenthesis 2 plus x right parenthesis over denominator x squared plus 3 x minus 10 end fraction
    Step 2 of 3:
    Simplify the denominator:
    fraction numerator left parenthesis 2 minus x right parenthesis left parenthesis 2 plus x right parenthesis over denominator x squared plus 3 x minus 10 end fraction equals fraction numerator left parenthesis 2 minus x right parenthesis left parenthesis 2 plus x right parenthesis over denominator x squared plus 5 x minus 2 x minus 10 end fraction
    equals fraction numerator left parenthesis 2 minus x right parenthesis left parenthesis 2 plus x right parenthesis over denominator x left parenthesis x plus 5 right parenthesis minus 2 left parenthesis x plus 5 right parenthesis end fraction
    equals fraction numerator left parenthesis 2 minus x right parenthesis left parenthesis 2 plus x right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x plus 5 right parenthesis end fraction
    equals fraction numerator negative left parenthesis x minus 2 right parenthesis left parenthesis x plus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x plus 5 right parenthesis end fraction
    equals fraction numerator negative left parenthesis x plus 2 right parenthesis over denominator left parenthesis x plus 5 right parenthesis end fraction
    Thus, the simplified expression is: fraction numerator negative left parenthesis x plus 2 right parenthesis over denominator left parenthesis x plus 5 right parenthesis end fraction
    Step 3 of 3:
    Here, the denominator of the expression cannot be zero. It is zero when:
    So, the domain of the expression is: left parenthesis negative straight infinity comma negative 5 right parenthesis union left parenthesis negative 5 comma straight infinity right parenthesis.

     
     

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