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Examples and theorems for finding integral and rational zeros of polynomial functions with integral coefficients. It includes the Integral Zeros Theorem and Rational Zeros Theorem, as well as examples for determining the integral and rational zeros of specific polynomials and writing the polynomials as the product of linear factors.
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7.3 Integral and Rational Zeros of Polynomials Integral Zeros Theorem: if an integer a is a zero of a polynomial function with integral coefficients and a leading coefficient of 1, then a is a factor of the constant term of the polynomial. Integral Zeros Consider: f(x) = x^3 + 4x^2 7x 10
Example 1 : Determine the integral zeros of f(x) = x^3 6x 2 + 3x + 10 and write f(x) as the product of linear factors.
Rational Zeros Theorem: Let f(x) = anxn^ + an1 xn1^ + an2 xn2^ + ... + a 1 x + a 0 , a 0 ≠ 0, be a polynomial function in standard form that has integral coefficients. Then if the nonzero rational number p/q in lowest terms is a zero of p(x), p must be a factor of the constant a 0 and q must be a factor of the leading coefficient an. Consider f(x) = 4x^3 + 7x^2 43x + 35
Example 3 : Determine the rational zeros of f(x) = 12x^3 16x 2 5x + 3 and write f(x) as the product of linear factors.
