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How do you factor #x^6-26x^3-27#?

Answer 1

Isaiah Anthony

First let's notice that #x^6=(x^3)^2#

Let #r=x^3#

Using substitution we have

#r^2-26r-27#

We now look for the factors of #-27# that will combine (add) to make #-26#

We find that #-27# and #1# multiply to be #-27# and add to be #-26#. This gives us

#(r-27)(r+1)#

Now substitute #x^3# back in for #r#

#(x^3-27)(x^3+1)#

We should recognize this as a difference of cubes and a sum of cubes.

#(x-3)(x^2+3x+9)(x+1)(x^2-x+1)#

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Answer 2

Isaiah Anthony

To factor the expression (x^6 - 26x^3 - 27), you can treat it as a quadratic in terms of (x^3). Let (y = x^3), then the expression becomes (y^2 - 26y - 27).

Now, factor the quadratic expression (y^2 - 26y - 27). The factors are ((y - 27)(y + 1)).

Substitute back (y = x^3): ((x^3 - 27)(x^3 + 1)).

Now, factor (x^3 - 27) using the difference of cubes formula, and (x^3 + 1) using the sum of cubes formula:

(x^3 - 27 = (x - 3)(x^2 + 3x + 9)), (x^3 + 1 = (x + 1)(x^2 - x + 1)).

Therefore, the factored form of (x^6 - 26x^3 - 27) is: ((x - 3)(x^2 + 3x + 9)(x + 1)(x^2 - x + 1)).

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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.