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 x x x ; g x x4 3 28 15 4 42. ( ) ( )= + - = -f x x x ; g x x23 5 45.5 The Real Zeros of a Polynomial FunctionRemainder Theorem: If f(x) is divided by x-c, then the remainder is f(x). Factor theorem: If f is a polynomialfunction, then x – c is a factor of f(x) if any only if f(c) = 0. Number of Real Zeros Theorem: A polynomial functioncannot 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 coefficientsof f(x) or else equals that number less an even integer; (b) ) the number of negative real zeros of f either equals thenumber 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 coefficientis 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 isa 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 throughan-1 or 1 + the coefficient with the largest absolute value (see pg. 367 for mathematical symbols). All the zeros of thepolynomial function will fall between ±M. Intermediate Value Theorem: If f is a polynomial function and a < b andf(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 – 5Find the real zeros of each polynomial and factor over the real numbers:3. f(x) = 2x6 + 3x5 – 3x4 + 10x3 - 24x2 + 3x + 9Max. Zeros: Positive Zeros: Negative Zeros: Potential Zeros: Real Zeros: 4. f(x) = 4x5 + 12x4 - x - 3Max. Zeros: Positive Zeros: Negative Zeros: Potential Zeros: Real Zeros:Solve the equation in the real number system:5.3 233 2 02x x x+ + - = 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:Professor P. Bishop MAC1147 (MDN)12. x5 - 2x3 - x2 - 2x – 7 = 0Try these (R.6, 5.5)R.61. f(x) = x5 + 1; g(x) = x + 12. Determine if x + 1/3 is a factor 3x4 + x3 – 3x + 13. Find the sum of a, b, c, and d if: 3 222 3 52 2x x x dax bx cx x- + += + + ++ +4.64. Solve: 2x4 + x3 – 24x2 + 20x + 16 = 0Max. Zeros: Positive Zeros: Negative Zeros: Potential Zeros: Real Zeros:5. Graph: f(x) = x4 – x3 – 6x2 + 4x +
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