Whether this statement is true or false

  • Thread starter an_mui
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In summary, the statement "x-1" is a factor of a polynomial in the form (x^n - 1) where 'n' is a positive integer is always true. This is because (x^n - 1) is a difference of squares that can be further factored into (x^(n/2) - 1)(x^(n/2) + 1), where the power of x is n/(2^k). When n is an odd integer, dividing by 2^k will never equal 1, but by using long division or the remainder theorem, it can be proven that (x-1) is still a factor. Additionally, when x=1, (x^n -
  • #1
an_mui
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'x - 1' is a factor of a polynomial in the form (x^n - 1) where 'n' is a positive integer.

my guess:

This statement is always true because (x^n - 1) is a difference of square. When factored even more, (x^n - 1) = (x^n/2 - 1)(x^n/2 + 1). Therefore, (x-1) can be factor of (x^n - 1) and is always true.
 
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  • #2
(x^n - 1) = (x^(n/2) -1) (x^(n/2) - 1)
= (x^(n/4) - 1)(x^(n/4) + 1)(x^(n/2) + 1)

and so on, where the power of x is n/(2^k).

Consider the case where n is odd, dividing n by 2^k will never equal 1.
 
  • #3
Your logic is faulty. Think about what your statement means when n is an odd integer and what is meant by factoring a polynomial.
 
  • #4
oops i think i know now

(x-1)(x^(n-1) + 1) =(x^n - 1), where n is a positive integer greater than or equal to 2
 
  • #5
Not quite! I suggest trying long division! If there is no remainder then x-1 is a factor. :)
 
  • #6
To help with the long division, you might want to try it in the special cases where n=2, 3, 4, ... or as many as needed before you see a pattern. Then try to prove this pattern works for a general n.

Or you can avoid long division by using the remainder theorem.

Or you can do both and be even more convinced.
 
  • #7
More simply, when x= 1, xn- 1 becomes 1-1= 0. Therefore, x- 1 is a factor (I just noticed that shmoe referred to the "remainder theorem"- that's what this is).
 

1. Is it possible for a statement to be both true and false?

No, a statement cannot be both true and false at the same time. It is either one or the other.

2. How do you determine if a statement is true or false?

The truth or falsity of a statement is determined by examining the evidence and logical reasoning. It can also depend on the context and perspective.

3. Can a statement be partially true and partially false?

Yes, a statement can contain elements that are true and elements that are false. This is known as a partly true or partly false statement.

4. What if there is not enough evidence to determine the truth or falsity of a statement?

In this case, the statement is considered to be inconclusive or uncertain. More evidence or further analysis may be needed to determine its truth or falsity.

5. Can a statement be true or false for one person but not for another?

Yes, a statement can be subjective and can be perceived differently by different people. This can be influenced by personal beliefs, biases, and experiences.

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