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How do you simplify #(x³-27 )/ (x-3)#?

Answer 1

Use the formula for the difference of two cubes to factor the top and cancel the #x-3# to get #x^2+3x+9#

The formula for the difference of two cubes is #a^{3}-b^{3}=(a-b)(a^2+ab+b^2)# (check this by expanding out the right side).

Thus, since #x^3-27=x^3-3^3#,

#(x^3-27)/(x-3)=((x-3)(x^2+3x+9))/(x-3)=((cancel(x-3))(x^2+3x+9))/(cancel(x-3))=x^2+3x+9#

This last equality is true as long as #x!=3#.

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Answer 2

Eva Abramson

To simplify the expression (x³-27)/(x-3), we can use the factorization of a difference of cubes. The numerator, x³-27, can be written as (x-3)(x²+3x+9). Therefore, the expression simplifies to (x-3)(x²+3x+9)/(x-3). The common factor of (x-3) cancels out, leaving us with the simplified expression of x²+3x+9.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.