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How do I factor polynomials and trinomials?

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If P(x)=a_nx^n+a_(n-1)x^(n-1)+cdots+a_1x+a_0, then you can factor P by finding solutions of algebraic equation   a_nx^n+a_(n-1)x^(n-1)+cdots+a_1x+a_0=0 which can sometimes be quite difficult.
There are formulas for solving equations of degree lower than 5, but there are no formula for solving algebraic equation of degree 5 and higher (it's not that formula has not been found, but it does not exist, it can never be found).
So if x_1, x_2, ldots, x_n are solutions to our equation then we can write our polynomial as: P(x)=(x-x_1)(x-x_2)cdots(x-x_n).
If k_1,k_2,ldots,k_m are multiplicities of x_1,x_2,ldots,x_m respectivly, then we can write P as: P(x)=(x-x_1)^(k_1)(x-x_2)^(k_2)cdots(x-x_m)^(k_m).
If you find only one root x_0 then you can write P as: P(x)=(x-x_0)Q(x). you can get polynomial Q    by deviding P with (x-x_0).

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How do I factor polynomials and trinomials?

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