Maths-
General
Easy

Question

How can you use polynomial identities to factor polynomials and simplify numerical expressions ? x cubed minus 216

Hint:

open parentheses a cubed minus b cubed close parentheses equals left parenthesis a minus b right parenthesis open parentheses a squared plus a b plus b squared close parentheses , where a and b can be real numbers, variables or multiples of both.
We are asked to explain on how the polynomial identities can be used to factorize polynomials and simplify them.

The correct answer is: = (x - 6)(x2 + 6x + 36)


     Step 1 of 2:
    Factoring a polynomial is the process of decomposing a polynomial into a product of two or more polynomials. This can be done using identities, which would speed up the process and make it simple.
    Here, the given expression is x3 - 216, which can be written as: x3 - 63
    Step 2 of 2:
    Apply the polynomial identity open parentheses a cubed minus b cubed close parentheses equals left parenthesis a minus b right parenthesis open parentheses a squared plus a b plus b squared close parentheses to factorize the expression x3 - 216:

    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 cubed minus 6 cubed equals left parenthesis x minus 6 right parenthesis open parentheses x squared plus 6 x plus 6 squared close parentheses end cell row cell equals left parenthesis x minus 6 right parenthesis open parentheses x squared plus 6 x plus 36 close parentheses end cell end table
     

    We use identities to speed up the process of multiplication and simplification. There are some basic polynomial identities that you need to by heart.

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