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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)=\(3x^{3}+6x^{2}-15x-30\)



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