222 CHAPTER 4 Polynomial and Rational Functions 5 Solve Polynomial Equations Solving a Polynomial Equation Solve the equation: x x x x x 4 16 37 84 84 0 6 5 3 2 + − − − − = EXAMPLE 7 Solution The solutions of this equation are the zeros of the polynomial function f x x x x x x 4 16 37 84 84 6 5 3 2 ( ) = + − − − − Using the result of Example 6, the real zeros are 2, 7, − − and 7.These are the real solutions of the equation x x x x x 4 16 37 84 84 0 6 5 3 2 + − − − − = . Now Work PROBLEM 69 In Example 6, the quadratic factor x 3 2 + that appears in the factored form of f x( ) is called irreducible, because the polynomial x 3 2 + cannot be factored over the real numbers. In general, a quadratic factor ax bx c 2 + + is irreducible if it cannot be factored over the real numbers, that is, if it is prime over the real numbers. Refer again to Examples 5 and 6. The polynomial function of Example 5 has three real zeros, and its factored form contains three linear factors. The polynomial function of Example 6 has three distinct real zeros, and its factored form contains three distinct linear factors and one irreducible quadratic factor. THEOREM Every polynomial function with real coefficients can be uniquely factored into a product of linear factors and/or irreducible quadratic factors. THEOREM A polynomial function of odd degree with real coefficients has at least one real zero. THEOREM Bounds on Zeros Let f denote a polynomial function whose leading coefficient is positive. • If M 0 > is a real number and if the third row in the process of synthetic division of f by x M− contains only numbers that are positive or zero, then M is an upper bound to the real zeros of f . • If m 0 < is a real number and if the third row in the process of synthetic division of f by x m− contains numbers that alternate positive (or 0) and negative (or 0), then m is a lower bound to the real zeros of f . COMMENT The bounds on the real zeros of a polynomial provide good choices for setting the viewing rectangle of a graphing utility. Within these bounds, all the x -intercepts of the graph can be seen. ■ We prove this result in Section 4.4, and in fact, we shall draw several additional conclusions about the zeros of a polynomial function. One conclusion is worth noting now. If a polynomial with real coefficients is of odd degree, it must have at least one linear factor. (Do you see why? Consider the end behavior of polynomial functions of odd degree.) This means that it must have at least one real zero. 6 Use the Theorem for Bounds on Zeros The work involved in finding the zeros of a polynomial function can be reduced somewhat if upper and lower bounds to the zeros can be found. A number M is an upper bound to the zeros of a polynomial f if no zero of f is greater than M . The number m is a lower bound if no zero of f is less than m . Accordingly, if m is a lower bound and M is an upper bound to the zeros of a polynomial function f , then m f M any zero of ≤ ≤ For polynomials with integer coefficients, knowing the values of a lower bound m and an upper bound M may enable you to eliminate some potential rational zeros– that is, any zeros outside the interval m M, [ ] .
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