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2.5 Zeros of Polynomial FunctionsThe Fundamental Theorem of Algebra – If f(x) is a polynomial of degree n, where n 0, then f has atleast one zero in the complex number system.Linear Factorization Theorem – If f(x) is a polynomial of degree n, where n 0, then f has precisely nlinear factors f x an x c1 x c2 . x cn where c1 , c2 ,., cn are complex numbers.Examples: Find all the zeros of the function. 1. f x x 2 x 3 x 2 12. f x x 6 x i x i nn 12The Rational Zero Test – If the polynomial f x an x an 1 x . a2 x a1 x a0 has integercoefficients, every rational zero of f has the form Rational Zero pwhere p and q have no commonqfactors other than 1, and p is a factor of a0 and q is a factor of an .

Examples: Find all the rational zeros of the function.31. f x x 7 x 6322. f x x 9 x 20 x 12323. c x 2 x 3x 1

Example: Find all real solutions of x4 13x2 12 x 032Example: List the possible rational zeros of f x 3x 20 x 36 x 16 . Sketch the graph of f sothat some of the possible zeros can be disregarded and then determine all real zeros of f.Complex Zeros Occur in Conjugate Pairs – Let f(x) be a polynomial function that has real coefficients. Ifa bi , where b 0 , is a zero of the function, the conjugate a bi is also a zero of the function.

Factors of a Polynomial – Every polynomial of degree n 0 with real coefficients can be written as theproduct of linear and quadratic factors with real coefficients, where the quadratic factors have no realzeros.Examples: Find a polynomial function with real coefficients that has the given zeros.1. 4, 3i2. 5,3, 2i3. 5, 5,1 3i4. You try it: 3, 4,5i, inn 12Descartes’s Rule of Signs – Let f x an x an 1 x . a2 x a1 x a0 be a polynomial with realcoefficients and a0 0 .1. The number of positive real zeros of 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 f is either equal to the number of variations in sign off x or less than that number by an even integer.

Examples: Use Descartes’s Rule of Signs to determine the possible numbers of positive and negativezeros of the function.21. h x 4 x 8x 3322. f x 5x x x 5Upper and Lower Bound Rules – Let f(x) be a polynomial with real coefficients and a positive leadingcoefficient. 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 for thereal zeros of f.2. If c 0 and the numbers in the last row are alternately positive and negative (zeros count aspositive or negative), c is a lower bound for the real zeros of f.

Examples: Verify the upper and lower bounds of the real zeros of f.321. f x x 3x 2 x 1a) Upper: x 1b) Lower: x 442. f x 2 x 8x 3a) Upper: x 3b) Lower: x 4