How do you factor #x^9 - 27#?
You can use the difference of cubes identity to help factor this, but to find all the factors with Real coefficients requires a little more...
The difference of cubes identity can be written:
The remaining cubic factor can itself be treated as a difference of cubes:
The remaining sextic factor can be factored as the product of three quadratics:
To see why requires some Complex arithmetic.
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To factor ( x^9 - 27 ), first, note that ( 27 = 3^3 ). Then, use the difference of cubes formula, which states that ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) ). Apply this formula with ( a = x^3 ) and ( b = 3 ) to factor ( x^9 - 27 ) as ( (x^3 - 3)(x^6 + 3x^3 + 9) ).
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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.
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.
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.
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.

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