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Professor P. Bishop MAC1147 (MDN) R.6 Synthetic Division (Condensed) Synthetic division is used to divide a polynomial of degree 1 or higher by a binomial g(x) of the form x – c. Use synthetic division to determine whether x – c is a factor of f(x): 1. f x x 4 x 3 8x 2 15x 4; g x x 4 2. f x x 2 3x 5; g x x 4 5.5 The Real Zeros of a Polynomial Function Remainder Theorem: If f(x) is divided by x-c, then the remainder is f(x). Factor theorem: If f is a polynomial function, then x – c is a factor of f(x) if any only if f(c) = 0. Number of Real Zeros Theorem: A polynomial function cannot have more real zeros than its degree. Descartes’ Rule of Signs: If f is a polynomial function in standard form, then (a) the number of positive real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f(x) or else equals that number less an even integer; (b) ) the number of negative real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f(-x) or else equals that number less an even integer. Rational Zero Theorem: Given f is a polynomial function of degree 1 or higher in descending order and each coefficient is an integer, if p/q, in lowest terms, is a rational zero of f, then q is a factor of the lead coefficient ( ≠0) and p is a factor is a factor of last coefficient (≠0). Bounds on Zeros: Let f be a polynomial function whose leading coefficient is 1: f(x) = xn + an-1xn-1 + … + a1x + a0. The bound M on the zeros of f is the smaller of the sum of the absolute values of a0 through an-1 or 1 + the coefficient with the largest absolute value (see pg. 367 for mathematical symbols). All the zeros of the polynomial function will fall between ±M. Intermediate Value Theorem: If f is a polynomial function and a < b and f(a) and f(b) have opposite signs, then there is a zero between a and b. Determine if g(x) is a factor of f(x) = x4 – 5x3 - 3x2 + 20x - 25 by finding f(c): 1. g(x) = x + 3 2. g(x) = x – 5 Find the real zeros of each polynomial and factor over the real numbers: 3. f(x) = 2x6 + 3x5 – 3x4 + 10x3 - 24x2 + 3x + 9 Max. Zeros: Positive Zeros: Negative Zeros: Potential Zeros: Real Zeros: 4. f(x) = 4x5 + 12x4 - x - 3 Max. Zeros: Positive Zeros: Potential Zeros: Real Zeros: Negative Zeros: Solve the equation in the real number system: 3 5. x 3 x 2 3 x 2 0 2 Max. Zeros: Positive Zeros: Negative Zeros: Potential Zeros: Real Zeros: Find a possible graph for the following: 6. f(x) = 4x5 + 12x4 - x - 3 7. f(x) = 2x6 + 3x5 – 3x4 + 10x3 - 24x2 + 3x + 9 Find the bound on the real zeros of the polynomial function: 8. f(x) = x4 – 5x2 – 36 9. f(x) = 3x4 – 3x3 - 5x2 + 27x – 36 Use the Intermediate Value Theorem to show that the polynomial has a zero in the given interval: 10. f(x) = 2x4 + x3 – 24x2 + 20x + 16; [0, -1) Given that the equation has a solution r in the given interval, approximate the solution correct to two decimal places: 11. x4 - 4x3 - x2 - 2x – 3 = 0 [4, 5] The following equation has exactly one positive zero. Approximate its value to two decimal places: 12. x5 - 2x3 - x2 - 2x – 7 = 0 Professor P. Bishop MAC1147 (MDN) Try these (R.6, 5.5) R.6 1. f(x) = x5 + 1; g(x) = x + 1 2. Determine if x + 1/3 is a factor 3x4 + x3 – 3x + 1 3. Find the sum of a, b, c, and d if: x3 2 x2 3 x 5 d ax 2 bx c x2 x2 4.6 4. Solve: 2x4 + x3 – 24x2 + 20x + 16 = 0 Max. Zeros: Potential Zeros: Real Zeros: 5. Positive Zeros: Graph: f(x) = x4 – x3 – 6x2 + 4x + 8 Negative Zeros: