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2.5 Zeros of Polynomial Functions We have known that a polynomial of degree n can have up to n zeros. Working with the complex number system now allows us to state that it has exactly n zeros when taking into account multiplicity. Find all the zeros of each function: f(x) = x - 4 f(x) = x2 - 4x + 4 = (x - 2)(x - 2) f(x) = x(x + 3i)(x - 3i) f(x) = (x + 2)(x - 2)(x + 2i)(x - 2i) The Rational Zero Test relates the possible rational zeros of a polynomial to the leading coefficient and to the constant term. The Rational Zero Test: If a polynomial f(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 has integer coefficient, every rational zero of f has the form p/q, where p and q have no common factors other than 1, and p is a factor of the constant term a0 and q is a factor of the leader coefficient an. Make a list of the possible rational zeros: f(x) = x3 - 7x - 6 f(x) = 4x3 - 19x2 + 33x - 9 1 Find all the rational zeros of f(x) = x3 + x + 1 Find all the rational zeros of f(x) = x3 - 5x2 + 2x + 8 f(x) = x3 - 15x2 + 75x - 125 f(x) = 2x4 - 9x3 - 18x2 + 71x - 30 2 Find the zeros of f(x) = 6x4 - 11x3 - 51x2 + 99x - 27 Complex Zeros Occur in Conjugate Pairs: If a + bi, where b ≠ 0, is a zero of a function, then a - bi is also a zero. Find a polynomial function that has 2, 7i, and -7i as zeros: Find a polynomial function that has 6, -5 + 2i, and -5 - 2i as zeros: 3 Find all the zeros of f(x) = 2x3 + 3x2 + 50x + 75 if 5i is one zero: Find all the zeros of f(x) = x3 - 4x2 + 21x - 34 if 1 + 4i is one zero: Find the zeros and write as a product of linear factors: g(x) = x2 + 10x + 23 h(x) = x3 - 11x2 + 41x - 51 4 Descartes's Rule of Signs: 1) The number of positive real zeros of a function f is either equal to the number of variations in sign of f(x) or less than that number by an even integer. 2) The number of negative real zeros of a function f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer. A variation in sign means that two consecutive coefficients have opposite signs. Count your multiplicities as multiple zeros. f(x) = -2x3 + 5x2 - x + 8 A value is called an upper bound of a function if no zeros are greater than that value. Similarly with a lower bound. Upper and Lower Bound Rules: Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x - c, using synthetic division: 1) If c > 0 and each number in the last row is either positive or zero, c is an upper bound of f. 2) If c < 0 and the numbers in the last row are alternately positive and negative (zero counts as either), c is a lower bound of f. Verify an upper bound of 4 and a lower bound of -1 for 2x4 - 8x + 3 5 Find all the real zeros of f(x) = 6x3 - 4x2 + 3x - 2 Find all the real zeros of f(x) = 8x3 - 4x2 + 6x - 3 6