How to Use the Rational Zeros Theorem in Algebra

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The rational zeros theorem provides a list of possible rational solutions for a polynomial equation of the form a_nx^n + a_n-1x^n-1 + ... +a_0 = 0 where a_i are integer coefficients and a_0 and a_n are nonzero. The following steps will show how to use the rational zeros theorem in algebra.

  • Express all rational solutions (roots) x for a polynomial equation as x = p/q where p and q are integers such that p is a factor of a_0 and q is a factor of a_n. All of the rational roots x for a polynomial can therefore be expressed as x = p/q or x = - p/q.

  • Solve a simple polynomial with the rational zeros theorem. For the polynomial x^3 - 4x + x + 2 we have a_n = 1 and a_0 = 2. The only integer factors for a_n is 1 and the factors for a_0 are 1 and 2.

  • Determine the possible solutions for the polynomial based on the factors obtained in Step 2. We have p = 1 or 2 and q = 1 and therefore x = p/q = 2/1 = 2, x = p/q = 1/1 = 1, x = -p/q = -(2/1) = -2 and x = -p/q = -(1/1) = -1. Possible solutions for the polynomial given in Step 2 are therefore -2, -1, 1 and 2.

  • Test the possible solutions given in Step 3 in the polynomial given in Step 2. x^3 - 4x + x + 2 = 0 when x = -2 and 1. By factoring (x+2) and (x-1) out of the equation, we see that the third root is also 1. Thus, x^3 - 4x + x + 2 = (x+2)(x-1)(x-1)=0 and the solutions are 1, 1 and 2.

  • Use the Horner scheme for more difficult solutions to the polynomial. If none of the possible solutions determined by the rational zeros theorem are actually solutions, the given polynomial has no rational solutions.

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