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Analysis BC Miss Hall Unit 5 Day 4: Irrational Zeros & IVT Complex Root Theorem: Given a polynomial function, f, if a + bi is a root of the polynomial then a – bi must also be a root. Example: Find a polynomial with rational coefficients with zeros 2, 1 + 3 , and 1 – i. Intermediate Value Theorem (IVT): Given real numbers a & b where a < b. If a polynomial function, f, is such that f(a) ≠ f(b) then in the interval [a, b] f takes on every value between f(a) to f(b). Examples: 1) First use your calculator to find the zeros of f ( x) 3 x 4 x3 2 x 2 5 . 8 Now verify the1 unit integral interval that the zeros are in using the Intermediate Value Theorem. 2) Use the Intermediate Value Theorem to find the 1 unit integral interval for each of the indicated number of zeros. a. One zero: g ( x) 3x3 4 x 2 x 3 b. Four zeros: f ( x) x 4 10 x 2 2 Analysis BC Miss Hall Unit 5 Day 4: Irrational Zeros & IVT 3) Given f ( x) 3x3 4 x 2 3x 2 : a. What is a value guaranteed to be between f(2) and f(3). b. What is another value guaranteed to be there? c. What is a value that is NOT guaranteed to be there? d. But could your value for c be there? Sketch a graph to demonstrate your answer. 4) Given a polynomial, g, where g(0) = -5 and g(3) = 15: a. True or False: There must be at least one zero to the polynomial. Explain. b. True or False: There must be an x value between 0 and 3 such that g(x) = 12. Explain. c. True or False: There can not be a value, c, between 0 and 3 such that g(c) = 25. Explain. Homework: 1) Find the equation of the quadratic whose only x-intercept is (-4, 0) and passes through (-2, 8) Analysis BC Miss Hall Unit 5 Day 4: Irrational Zeros & IVT 2) Graph y = -½ x2 + ½ x + 3. 3) Given f ( x) x6 6 x5 8 x 4 14 x3 45 x 2 40 x 12 a. What is the end behavior b. List the possible rational roots 4) Divide: (2 x5 8 x3 2 x 2 4 x 2) (2 x 4) 5) Use the remainder theorem to find f(5) for f(x) from problem #3. 6) Use your calculator to approximate the real zeros and relative extrema of the following functions. a. f ( x) 2 x 4 6 x 2 1 b. f ( x) x 5 3 x 3 x 6 Analysis BC Miss Hall Unit 5 Day 4: Irrational Zeros & IVT 7) Find a polynomial function that has the given zeros. a. 0, 4 b. 0, -2, -3 c. 4, -3, 3, 0 d. 1 + 3 e. 2, 4 – 5 8) For each of the following: (a) use the IVT to find integral intervals one in length which must contain a zero (b) now use your calculator to find the zeros (checking your answer to part (a). a. f ( x) x3 3x 2 3 b. g ( x) 3x 4 4 x3 3 Analysis BC Miss Hall Unit 5 Day 4: Irrational Zeros & IVT 9) For each of the following: Identify the symmetry it has (x-axis, y-axis or origin), and determine the number of x-intercepts it has. Use your calculator to verify your answer. a. f ( x) x 2 ( x 6) 1 b. g (t ) (t 4) 2 (t 4) 2 2 c. f ( x) x 3 4 x d. 1 f ( x) ( x 1) 2 ( x 3)(2 x 9) 5 10) An open box is to be made from a square piece of material 36cm on a side by cutting equal squares with sides of length x from the corners and turning up the sides. a. Find an equation for the volume of the box, V(x) b. Determine the domain of the function V c. Use your calculator to find the length, x, for which the maximum volume is produced. Analysis BC Miss Hall Unit 5 Day 4: Irrational Zeros & IVT Answers: 1) y = 2x2 + 16x + 32 2) check with calc 63 4) x 2 x 8 x 15 x 32 x2 4 3 2 3) a) up, up b) 12, 6, 4, 3, 2, 1 zeros : 1.680, 0.421 5) 2688 6) a) Re l Max :(0,1) Abs Min : (1.225, 3.500) zeros : 1.178 b) Re l Max :(0.324, 6.218) Re l Min : (0.324,5.782) 7) a) f ( x) x 2 4 x b) f ( x) x3 5x 2 6 x c) f ( x) x 4 4 x3 9 x 2 36 x d) f ( x) x 2 2 x 2 e) f ( x) x3 10 x 2 27 x 22 8) a) (-1,0) (1, 2) (2, 3): -0.879, 1.347, 2.532 9) check with calc b) (-2, -1) (0,1): -1.585, 0.779 10) a) V(x) = x(36 – 2x)2 b) Domain (0, 18) c) 6cm