The factorial theorem stipulates that if the polynomial f(a)=0, the polynomial f(x) must contain the factor x-a; conversely, if the polynomial f(x) contains the factor x-a, the polynomial f(a)=0.

Steps to decompose factors by factorial theorem.

1. Guess the root of polynomial f(x) according to the coefficients of the highest term and constant term.

2. Substitute the guess value A into polynomial verification,

If f(a)≠0, the guess is wrong, and f(x) does not contain the factor (x-a). Try another root.

If f(a)=0, the guess is correct. According to the factorial theorem, f(x) contains factorial (x-a), and then proceed to the next step.

3. Divide f(x)/(x-a)=g(x) by polynomial to get quotient polynomial g(x).

4. With 1, guess and verify the root of g(x), and use factorial theorem to find a factor of g(x).

Repeat the above process until the quotient term becomes linear or indecomposable.

Use f(x)=3x? Take +x-2 as an example. If f(x) is decomposable, then its rational roots must be in 1, 2, 1/3, 2/3. There is no need to try one by one. Through observation (mental arithmetic), it is known that f(- 1)=0.

Another example is: -x 4+2x 3+4x 2- 10x+5 = 0.

That is: x 4-2x 3-4x 2+ 10x-5 = 0.

Observation (mental arithmetic) shows that X 4-2x 3-4x 2+ 10x-5 contains the factor x- 1.

So: (x 3-x 2-5x+5) (x- 1) = 0.

Further observation shows that x 3-x 2-5x+5 also contains the factor x- 1.

So: (x 2-5) (x-1) (x-1) = 0.

Therefore, the solution of the equation -x 4+2x 3+4x 2- 1x+5 = 0 can be obtained as follows:

X 1=x2= 1, x3= radical number 5, x4=- radical number 5.