How can -x^3 + 27 be factored?

AI Thread Summary
To factor -x^3 + 27, the relevant formula is a^3 - b^3 = (a - b)(a^2 + ab + b^2). Here, 27 can be expressed as 3^3, allowing the expression to be rewritten as -(x^3 - 27). This fits the formula, where a = 3 and b = x, leading to the factorization - (x - 3)(x^2 + 3x + 9). The discussion highlights the importance of recognizing the structure of the expression to apply the correct factoring formula. Ultimately, the factorization process was clarified through collaborative problem-solving.
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Homework Statement


factor -x^3 + 27


Homework Equations





The Attempt at a Solution



Nothing, I have no idea how to factor it.
 
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Here are a couple of formulas for you:
a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
 
Mark44 said:
Here are a couple of formulas for you:
a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)

I tried those. My main problem is the -x^3.
 
You've got 3^3-x^3. That fits the second formula Mark44 gave you.
 
2 seconds on physics forum and I found what I'm after. Thanks very much.
 
Sorry we took so long!
 

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I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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