Maths-
General
Easy

Question

Use polynomial identities to factor the polynomials or simplify the expressions :
8 x cubed plus y to the power of 9

The correct answer is: Thus, the factor is: 8x^3 + y^9 = (2x+y^3) (4x^2 - 2xy^3 + y^6)


    ANSWER:
    Hint:
    open parentheses straight a cubed plus straight b cubed close parentheses equals left parenthesis straight a plus straight b right parenthesis open parentheses straight a squared minus ab plus straight b squared close parentheses, where a and b can be real values, variables or multiples of both.
    We are asked to use polynomial identities to factorize or simplify the expression.
    Step 1 of 2:
    The given expression is 8 x cubed plus y to the power of 9text  It can be written as  end text left parenthesis 2 x right parenthesis cubed plus open parentheses y cubed close parentheses cubed text . It is of the form  end text open parentheses a cubed plus b cubed close parentheses text  where  end text a equals 2 x straight & b equals y cubed text .  end text
    Step 2 of 2:
    Use the polynomial identity open parentheses straight a cubed plus straight b cubed close parentheses equals left parenthesis straight a plus straight b right parenthesis open parentheses straight a squared minus ab plus straight b squared close parentheses to simplify the expression;
    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 8 x cubed plus y to the power of 9 equals left parenthesis 2 x right parenthesis cubed plus open parentheses y cubed close parentheses cubed space space space space space space space space space space space space space space space space space space space space end cell row cell equals open parentheses 2 x plus y cubed close parentheses open parentheses left parenthesis 2 x right parenthesis squared minus left parenthesis 2 x right parenthesis open parentheses y cubed close parentheses plus open parentheses y cubed close parentheses squared close parentheses end cell row cell equals open parentheses 2 x plus y cubed close parentheses open parentheses 4 x squared minus 2 x y cubed plus y to the power of 6 close parentheses space space space space space space space space space space space space space end cell end table
    Thus, the factor is: 8 straight x cubed plus straight y to the power of 9 equals open parentheses 2 straight x plus straight y cubed close parentheses open parentheses 4 straight x squared minus 2 xy cubed plus straight y to the power of 6 close parentheses
    Note:
    Polynomial identities are equations that are true for all possible values of the variable. We can perform polynomial multiplication by applying the distributive property to the multiplication of polynomials.

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