Sunday

December 21, 2014

December 21, 2014

Posted by **Jess** on Wednesday, December 5, 2012 at 11:43am.

write p(x) as a product of irreducible polynomials.

- maths -
**Count Iblis**, Wednesday, December 5, 2012 at 3:42pmMod 5 there are only 5 numbers to try, zero is not a solution, so you only have to try ±1 and ±2. A more formal way would be to try some manipulations. If you write the constant term 1 as -4, you can divide the polynomial by 2:

[-4 + 4 x + 2 x^2 + 2 x^3]/2 =

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

Then since x is not zero, we have by Fermat's little theorem that x^4 = 1. So, you can replace the -2 by -2 x^4. If you then divide by x, (allowed snce x is not zero), you get:

-2 x^3 + 2 + x + x^2

Using this trick again by replacing the 2 by 2 x^4 and dividing by x, you get:

-2 x^2 + 2 x^3 + 1 + x

If you multiply this by 2 and add this to (1) you get

4 x -3 x^2 + 5 x^3

Then dividing by x and using that

5 = 0, you get:

4 - 3 x

Equating this to zero yields

x = 4*3^(-1) = 8 = 3

To factor the polynomial you thus need to divide it by (x-3) in Z5 (just use long division and do the calculations mod 5).

**Answer this Question**

**Related Questions**

Math - Roots Ok, what about roots? Roots of polynomials? Square roots? Cube ...

pre calc - A polynomial f(x) with real coefficients and leading coefficient 1 ...

maths - find the roots of the equation X(square)+7x equal to 0.(b)write down the...

maths - Find the product of roots of the following equation (4x + 7 )^2 =4x^2

maths - Find the value of k if the product of the roots of the quadratic ...

maths - Find the sum and product of the roots of the following equation. x^2 - ...

Polynomials - Steve it is a polynomials product it is asking me for the sum the ...

Algebra - Given the roots, 1/2 and 4, find: (A) The quadratic equation 2x^2-9x+4...

Algebra 2 - How do I solve polynomial equation by finding all complex roots? The...

further mathematics - The roots of the eqn, x^4 + px^3 + qx^2 + rx + s = 0 where...