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Zeros of Polynomial Functions Advanced Math Section 3.4 Number of zeros • Any nth degree polynomial can have at most n real zeros • Using complex numbers, every nth degree polynomial has precisely n zeros (real or imaginary) Advanced Math 3.4 - 3.5 2 Fundamental Theorem of Algebra • If f(x) is a polynomial of degree n, where n > 0, • then f has at least one zero in the complex number system Advanced Math 3.4 - 3.5 3 Linear Factorization Theorem • If f(x) is a polynomial of degree n, where n > 0, • then f has precisely n linear factors f x an x c1 x c2 where c1 , c2 , x cn cn are complex numbers Advanced Math 3.4 - 3.5 4 Linear Factorization Theorem applied 1st degree: f x x 5 has exactly one zero x -5 2nd degree: f x x 2 10 x 25 has exactly two zeros f x x 5 x 5 x 5 and x 5 (multiplicity counts: 5 is a repeated zero) 3rd degree: f x x3 +9x has exactly three zeros f x x x 2 9 x x 3i x 3i x 0, x 3i Advanced Math 3.4 - 3.5 5 Example • Find all zeros x4 1 Advanced Math 3.4 - 3.5 6 Rational Zero Test If the polynomial f x an x n an 1 x n 1 a2 x 2 a1 x a0 has integer coefficients, every rational zero of f has the form p rational zero q where p and q have no common factors other than 1, and p a factor of the constant term a0 q a factor of the leading coefficient an Advanced Math 3.4 - 3.5 7 Using the rational zero test • List all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient possible rational zeros = factors of constant term factors of leading coefficent • Use trial-and-error to determine which, if any are actual zeros of the polynomial • Can use table on graphing calculator to speed up calculations Advanced Math 3.4 - 3.5 8 Example • Use the Rational Zero Test to find the rational zeros f x x 4x 4x 16 3 2 Advanced Math 3.4 - 3.5 9 Using synthetic division • Test all factors to see if the remainder is zero • Can also use graphing calculator to estimate zeros, then only check possibilities near your estimate f x x 8x 40 x 525 3 2 Advanced Math 3.4 - 3.5 10 Examples • Find all rational zeros f x 2x3 3x2 8x 3 f x 2 x3 3x 2 1 Advanced Math 3.4 - 3.5 11 Conjugate pairs a bi a bi • If the polynomial has real coefficients, • then zeros occur in conjugate pairs • If a + bi is a zero, then a – bi also is a zero. Advanced Math 3.4 - 3.5 12 Example: • Find a fourth-degree polynomial function with real coefficients that has zeros -2, -2, and 4i Advanced Math 3.4 - 3.5 13 Factors of a Polynomial • Even if you don’t want to use complex numbers • Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros Advanced Math 3.4 - 3.5 14 Quadratic factors • If they can’t be factored farther without using complex numbers, they are irreducible over the reals x 1 x i x i 2 Advanced Math 3.4 - 3.5 15 Quadratic factors • If they can’t be factored farther without using irrational numbers, they are irreducible over the rationals – These are reducible over the reals x 2 x 2 x 2 2 Advanced Math 3.4 - 3.5 16 Finding zeros of a polynomial function • If given a complex factor – Its conjugate must be a factor – Multiply the two conjugates – this will give you a real zero – Use long division or synthetic division to find more factors • If not given any factors – Use the rational zero test to find rational zeros – Factor or use the quadratic formula to find the rest Advanced Math 3.4 - 3.5 17 Examples • Use the given zero to find all zeros of the function f x x x 9x 9, zero 3i 3 2 f x 4 x 23x 34 x 10, zero 3 i 3 2 Advanced Math 3.4 - 3.5 18 Examples • Find all the zeros of the function and write the polynomial as a product of linear factors h x x2 4 x 1 g x x3 6x2 13x 10 f x x4 10 x2 9 Advanced Math 3.4 - 3.5 19 Descartes’s Rule of Signs – A variation in sign means that two consecutive coefficients have opposite signs • For a polynomial with real coefficients and a constant term, • 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 • The number of negative real zeros is either equal to the variations in sign of f(-x) or less than that number by an even integer. Advanced Math 3.4 - 3.5 20 Examples • Determine the possible numbers of positive and negative zeros g x 2 x 3x 1 3 2 f x 3x 2 x x 3 3 2 Advanced Math 3.4 - 3.5 21 Upper Bound Rule – When using synthetic division • If what you try isn’t a factor, but • The number on the outside of the synthetic division is positive – And each number in the answer is either positive or zero – then the number on the outside is an upper bound for the real zeros Advanced Math 3.4 - 3.5 22 Lower Bound Rule – When using synthetic division • If what you try isn’t a factor, but • The number on the outside of the synthetic division is negative – The numbers in the answer are alternately positive and negative (zeros can count as either) – then the number on the outside is a lower bound for the real zeros Advanced Math 3.4 - 3.5 23 Examples • Use synthetic division to verify the upper and lower bounds of the real zeros f x x 4 4 x 3 15 f x 2 x3 3x 2 12 x 8 Upper : x 4 Upper : x 4 Lower : x 1 Lower : x 3 Advanced Math 3.4 - 3.5 24 Mathematical Modeling and Variation Advanced Math Section 3.5 Two basic types of linear models • y-intercept is nonzero y mx b • y-intercept is zero y mx Advanced Math 3.4 - 3.5 26 Direct Variation y kx for some nonzero constant k • • • • Linear k is slope y varies directly as x y is directly proportional to x Advanced Math 3.4 - 3.5 27 Direct Variation as an nth power y kxn for some constant k • y varies directly as the nth power of x • y is directly proportional to the nth power of x Advanced Math 3.4 - 3.5 28 Inverse Variation k y for some constant k x • Hyperbola (when k is nonzero) • y varies inversely as x • y is inversely proportional to x Advanced Math 3.4 - 3.5 29 Inverse Variation as an nth power k y n for some constant k x • y varies inversely as the nth power of x • y is inversely proportional to the nth power of x Advanced Math 3.4 - 3.5 30 Joint Variation z kxy for some constant k • Describes two different direct variations • z varies jointly as x and y • z is jointly proportional to x and y Advanced Math 3.4 - 3.5 31 Joint Variation as an nth and mth power z kx n ym for some constant k • z varies jointly as the nth power of x and the mth power of y • z is jointly proportional to the nth power of x and the mth power of y Advanced Math 3.4 - 3.5 32 Examples • Find a math model representing the following statements and find the constants of proportionality • A varies directly as r2. – When r = 3, A = 9p • y varies inversely as x – When x = 25, y = 3 • z varies jointly as x and y – When x = 4 and y = 8, z = 64 Advanced Math 3.4 - 3.5 33