Maths-
General
Easy

Question

How could you use polynomial identities to factor the expression x to the power of 6 minus y to the power of 6 .

The correct answer is: We can use multiple identities to simplify or factorize a polynomial fraction.


    ANSWER:
    Hint:
    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 open parentheses a squared minus b squared close parentheses equals left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis space space space space space space space space end cell row cell a cubed minus b cubed equals left parenthesis a minus b right parenthesis open parentheses a squared plus a b plus b squared close parentheses end cell row cell a cubed plus b cubed equals left parenthesis a plus b right parenthesis open parentheses a squared minus a b plus b squared close parentheses end cell end table
    Here, a and b can be real numbers, variables or multiples of both.
    We are asked to explain on how to use polynomial identities to factorize the given expression.
    Step 1 of 3:
    Factorization of algebraic means to obtain two or more expressions whose product is the given expression. The process of finding two or more expressions whose product is the given expression is called the factorization of algebraic expressions.
    Step 2 of 3:
    text  The given expression is  end text x to the power of 6 minus y to the power of 6 text . It can be written as  end text x to the power of 6 minus y to the power of 6 equals open parentheses x cubed close parentheses squared minus open parentheses y cubed close parentheses squared text . Using the identity, we have:  end text
    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 to the power of 6 minus y to the power of 6 equals open parentheses x cubed close parentheses squared minus open parentheses y cubed close parentheses squared end cell row cell equals open parentheses x cubed minus y cubed close parentheses open parentheses x cubed plus y cubed close parentheses end cell end table
    Step 3 of 3:
    Now, apply the identity of cubes to the simplified expression. Thus, we get:
    open parentheses straight x cubed minus straight y cubed close parentheses open parentheses straight x cubed plus straight y cubed close parentheses equals left parenthesis straight x minus straight y right parenthesis open parentheses straight x squared plus xy plus straight y squared close parentheses left parenthesis straight x plus straight y right parenthesis open parentheses straight x squared minus xy plus straight y squared close parentheses
    Thus, the factorized form is:
    open parentheses straight x cubed minus straight y cubed close parentheses open parentheses straight x cubed plus straight y cubed close parentheses equals left parenthesis straight x minus straight y right parenthesis open parentheses straight x squared plus xy plus straight y squared close parentheses left parenthesis straight x plus straight y right parenthesis open parentheses straight x squared minus xy plus straight y squared close parentheses
    Note:
    We can use multiple identities to simplify or factorize a polynomial fraction.

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