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MATHMAA

# Factorization Of Polynomials

Rational Zero Theorem :

Suppose f(x)=$$a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots +a_{0}$$ be a polynomial of degree n, $$a_{n}\neq 0$$  with integer coefficients. If $$a_{0},a_{n}$$ are non zero , then each rational solution x , when written as a fraction x=p/q in lowest terms satisfies

• p is integer factor of constant term $$a_{0}$$ and
• q is integer factor of leading coefficient $$a_{n}$$

Remarks:

1. Sum of all coefficients of a polynomial is 0 then x-1 is a factor .
2. Sum of the coefficients of even powers of x = sum of the coefficients of  odd powers of x , then x+1 is a factor .

Use the rational zeros theorem to find all the real zeros of the polynomial function. use the zeros to factor f(x) over the real numbers.

1)f(x)=$$3x^{3}+6x^{2}-15x-30$$