Factorisation of x^22 -3x^11 +2

  • Thread starter catcherintherye
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In summary, the given polynomial x^{22} -3x^{11} + 2 can be factored into p(x) = (x^11 -2)(x^11 - 1), where (x^11 -2) satisfies Eisenstein's criteria and x-1 is a factor of the second factor. Long division can be used to show that the second factor, x^{10} + x^9 +...+x + 1, is also irreducible. This can also be proven by substituting y=x+1 and applying Eisenstein's criteria.
  • #1
catcherintherye
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factorize [tex] x^{22} -3x^{11} + 2 [/tex]

right so I have p(x) = (x^11 -2)(x^11 - 1)

(x^11 -2) satifies eisentstein, obviously x-1 is a factor of the second factor. Long division reaps [tex] x^{10} + x^9 +...+x + 1 [/tex]

the solution asserts that this is also irreducible, but I do not see this?? Is this one of those where you substitue x for y+1??
 
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  • #2
Formatting tip: use curly braces {} to group the entire exponent together as one unit.
 
  • #3
Yea, any polynomial of the form (x^p-1)/(x-1) can be shown to be irreducible by making the substitution y=x+1 and applying Eisenstein's criteria. You have to do a little work with binomial coefficients, and it's probably easier to prove this general case then the case p=11.
 

What is factorisation?

Factorisation is the process of breaking down a mathematical expression into smaller parts or factors that can be multiplied together to get the original expression. It is a useful technique in solving equations and simplifying complicated expressions.

Why is factorisation important?

Factorisation allows us to simplify expressions and equations, making them easier to solve. It also helps us identify common factors and patterns, which can be useful in further mathematical calculations.

What is the general method for factorising a polynomial?

The general method for factorising a polynomial involves finding common factors and grouping terms. For a polynomial in the form of ax^2 + bx + c, we first check if there are any common factors among the coefficients a, b, and c. Then, we try to find two numbers that multiply to give us the constant term c, and add up to give us the coefficient b. These numbers can then be used to group the terms and factorise the polynomial.

How do we factorise x^22 -3x^11 +2?

To factorise x^22 -3x^11 +2, we first check for any common factors. In this case, x is a common factor, so we can factorise out x to get x(x^21 - 3x^10 + 2). Then, we can try to find two numbers that multiply to give us 2 and add up to give us -3. These numbers are -1 and -2. So, we can rewrite the expression as x(x^21 -x^10 -2x^11 +2) and factorise by grouping to get x(x^10(x^11 -1) -2(x^11 -1)). Finally, we can factorise out the common term (x^11 -1) to get the final factorised form of x(x^11 -1)(x^10 -2).

What are some applications of factorisation in real life?

Factorisation has various applications in real life, such as in cryptography, where it is used to break down large numbers into their prime factors to encrypt data. It is also used in engineering and physics to simplify complicated equations and models. In finance, factorisation is used in calculating interest rates and loan payments. It is also used in computer programming and data analysis to identify common factors and patterns in large datasets.

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