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November 18, 2013 Precalc Warm Up # 11-1 2 1. Divide x3 + 5x + 2x - 8 by x + 2. Was it a factor? 2. f(x) = 2x4+7x3-4x2-27x-18 a) Is (x-5) a likely factor of f(x)? b) Is (3x+2) a likely factor of f(x)? c) Is (x-2) a likely factor of f(x)? d) Factor f(x) completely November 18, 2013 2 1. Divide x3 + 5x + 2x - 8 by x + 2. Was it a factor? November 18, 2013 2. f(x) = 2x4+7x3-4x2-27x-18 a) b) c) d) Is (x-5) a likely factor of f(x)? Is (3x+2) a likely factor of f(x)? Is (x-2) a likely factor of f(x)? Factor f(x) completely November 18, 2013 Questions on #10-5? Divide using long division. November 18, 2013 Divide using synthetic division November 18, 2013 Use synthetic division to show that x is a solution of the third degree polynomial equation, and factor completely. November 18, 2013 November 18, 2013 November 18, 2013 November 18, 2013 DESCARTES'S RULE OF SIGNS: 1. the number of POSITIVE REAL ZEROs is either equal to the number of variations in sign of f(x) or less than that by an even integer. 2. the number of NEGATIVE REAL ZEROS is either equal to the number of variation in signs of f(-x) or less than that by an even integer. f(x) = 2x4+7x3-4x2-27x-18 had four zeros: 2,-3, -1, and -3/2 f(x) has 1 change in sign, which indicates that it has 1 positive real zero. f(-x) = 2(-x)4 + 7(-x)3 -4(-x)2-27(-x) - 18 = 2x4 - 7x3 - 4x2 + 27x - 18 therefore f(-x) has 3 changes in sign, indicating that it has 3 or 1 negative real zeros. November 18, 2013 HOW CAN THIS HELP US SOLVE HIGHER DEGREE EQUATIONS? Use Descartes's rule of signs to find all solutions to x4-3x3+x2+3x-2 = 0 November 18, 2013 We have also been taking advantage of the rational zero test Rational Zero Test: If f(x) = anxn + an-1xn-1 + ...+a1x + a0 has integer coefficients, then every rational zero of f is p/q, where p is a factor of constant term a0 and q is a factor of the leading coefficient an That looks more complicated than it is! It is just observing that if f(x) = 2x3+ 4x -5, then (3x+7) isn't a good factor guess. Which means -7/3 isn't going to be a zero. November 18, 2013 List all of the possible rational zeros of f. Using those and Descartes's rule of signs, find all of zeros and graph f(x)= -3x3+20x2-36x+16 November 18, 2013 List all of the possible rational zeros of f. Using those and Descartes's rule of signs, find all of zeros and graph f(x)= -3x3+20x2-36x+16 November 18, 2013 One last trick to have up your sleeve...... LOWER AND UPPER BOUND RULE 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, then c is an UPPER BOUND for the zeros 2. If c<0 and the numbers in the last row are alternately positive and negative (0 counts either way), then c is a LOWER BOUND for the zeros November 18, 2013 Let's use this on finding the real zeros of f(x) = 3x3-2x2+15x-10 3 variations of signs, so either 3 or 1 positive real zeros f(-x) = -3x3-2x2-15x-10 No variations of signs, so no negative real zeros Try 1. If it is an upper bound, what does this tell us? November 18, 2013 #11-1 p. 211 box, 53 The test is this Friday, and it will cover 3-1 --->3-6