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General
Easy

Question

What is the product of fraction numerator x plus 2 over denominator x to the power of 4 minus 16 end fraction straight & x cubed plus 4 x squared minus 12 x.

hintHint:

Simplify means to make it simple. In mathematics, simplify is the reduction of an expression/fraction into irreducible forms. The product of two expressions is the result of multiplying them.
We are asked to find the product of the expressions.

The correct answer is: Hence, the product is:


    Step 1 of 2:
    Simplify the term, fraction numerator x plus 2 over denominator x to the power of 4 minus 16 end fraction. Use the identity open parentheses x squared minus a squared close parentheses equals left parenthesis x plus a right parenthesis left parenthesis x minus a right parenthesis for simplification.
    fraction numerator x plus 2 over denominator x to the power of 4 minus 16 end fraction equals fraction numerator x plus 2 over denominator open parentheses x squared close parentheses squared minus 4 squared end fraction
    equals fraction numerator x plus 2 over denominator open parentheses x squared minus 4 close parentheses open parentheses x squared plus 4 close parentheses end fraction
    equals fraction numerator x plus 2 over denominator open parentheses x squared minus 2 squared close parentheses left parenthesis x plus 4 right parenthesis end fraction
    equals fraction numerator x plus 2 over denominator left parenthesis x minus 2 right parenthesis left parenthesis x plus 2 right parenthesis open parentheses x squared plus 4 close parentheses end fraction
    equals fraction numerator 1 over denominator left parenthesis x minus 2 right parenthesis open parentheses x squared plus 4 close parentheses end fraction
    Now, simplify the expressionx cubed plus 4 x squared minus 12 x using left parenthesis x plus a right parenthesis left parenthesis x plus b right parenthesis equals x squared plus left parenthesis a plus b right parenthesis x plus a b
    x cubed plus 4 x squared minus 12 x equals x open parentheses x squared plus 4 x minus 12 close parentheses
    equals x open parentheses x squared plus 6 x minus 2 x minus 12 close parentheses
    equals x left parenthesis x left parenthesis x plus 6 right parenthesis minus 2 left parenthesis x plus 6 right parenthesis right parenthesis
    equals x left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis
    equals x left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis
    Step 2 of 2:
    Now, find the product of the simplified expressions;
    fraction numerator x plus 2 over denominator x to the power of 4 minus 16 end fraction cross times x cubed plus 4 x squared minus 12 x equals fraction numerator 1 over denominator left parenthesis x minus 2 right parenthesis open parentheses x squared plus 4 close parentheses end fraction cross times space x left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis
    equals fraction numerator x left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis over denominator left parenthesis x minus 2 right parenthesis open parentheses x squared plus 4 close parentheses end fraction
    equals fraction numerator x left parenthesis x plus 6 right parenthesis over denominator open parentheses x squared plus 4 close parentheses end fraction
    Hence, the product is:       fraction numerator x left parenthesis x plus 6 right parenthesis over denominator open parentheses x squared plus 4 close parentheses end fraction


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    You could simplify an expression by removing common values form the numerator and the denominator.

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