Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a) What is the volume of the box if x = inches? b) What values of x make sense in this situation? Express the volume of the box as a function of x. c) What value(s) of x would yield the greatest volume? Int. Algebra Chapter 7 Summary page 1 of 14
. a) Let f(x) = 3x x +. Find f(). b) Let f(x) = x 3 + 1. Find f(-1). c) Let f(x) = x 3 + x +. Find f(a + 1). 3. Let f(x) = x 3 + x + 1, g(x) = x 3, find: a) (f + g)(x) = b) (f g)(x) = c) (f g)(x) = 4. Factor the polynomials: a) x 3 5x 6x b) x 16 c) x 3 + 4x + x + 8 d) x 3 + 8 Use sum of cubes: a 3 + b 3 = (a + b)(a ab + b ) e) 64x 3 1 Use difference of cubes: a 3 b 3 = (a b)(a + ab + b ) Int. Algebra Chapter 7 Summary page of 14
5. Use long division to find the quotient and the remainder when f(x) is divided by g(x). a) f(x) = 3x x + ; g(x) = x b) f(x) = x 3 + 1; g(x) = x + 1 c) f(x) = 5x 4 3x + ; g(x) = x + x + 1 Int. Algebra Chapter 7 Summary page 3 of 14
6. Suppose the result of dividing one polynomial by another is x 6x + 9 might have been divided? 1, what two polynomials x 3 7. What is the remainder when x 10 x + 4 is divided by x 1? Remainder Theorem: If a polynomial P(x) is divided by x a, then the remainder is. Proof: 8. Let f(x) = x 99 3 and d(x) = x + 1. Find the remainder when f(x) is divided by d(x). Factor Theorem: A polynomial function P(x) has a factor x a if and only if. Proof: Using the Remainder Theorem: P(x) = (x a) q(x) + r(x) Int. Algebra Chapter 7 Summary page 4 of 14
Fundamental Connections for Polynomial Functions: For a polynomial function f and a real number c, the following statements are equivalent: x = c is a solution (or root) of the equation f(c) = 0. c is a of the function f. c is an of the graph of y = f(x). x c is a of f(x). 9) Show that x + 3 is a factor of f(x) = x 3 + 6x x 30. Find the remaining factors. 10) Show that x 4 is a factor of f(x) = x 3 64. Find the remaining factor(s). Int. Algebra Chapter 7 Summary page 5 of 14
GRAPHS OF POLYNOMIAL FUNCTIONS 1. Graphs of monomial functions: f(x) = k or f(x) = kx n, where k = constant, n = positive integer. a) Graph the following functions on the same set of axes. Note the similarities and differences among the graphs. y = x, y = x 4, y = x 6 - b) Graph the following functions on the same set of axes. Note the similarities and differences among the graphs. y = x, y = x 3, y = x 5 - End behaviors a n > 0 a n < 0 Even degree Odd degree Int. Algebra Chapter 7 Summary page 6 of 14
MYSTERY FUNCTIONS 1. What function matches the given graph? Explain how you determine the function. a) b) 6 5 4 (3, 4) 5 c) d) 6 4 5 5 4 e) f) 5 4 80 5 6 70 60 50 g) h) 40 30 80 4 0 10 70 60 80 15 10 5 5 10 15 70 10 50 60 0 40 50 i) 40 30 30 j) There is only one zero at (-, 0) and y-intercept is (0, 16). 0 0 10 10 5 5 10 15 15 10 5 5 10 15 10 10 0 0 Int. Algebra Chapter 7 Summary page 7 of 14
k) 7 6 5 4 (-1, 3) 3 1 4 1 l) 3 1 1 3 4 State your conjectures about the number of zeros and the number of turning points and the general shape of the graph of a polynomial as it relates to the degree of the polynomial. How does the multiplicity of a zero affect the graph near that zero? Int. Algebra Chapter 7 Summary page 8 of 14
. Graph the polynomial function, locate its zeros, describe its end behaviors and explain how it relates to the monomial from which it is built. Use transformations whenever possible. a) f(x) = -x 3 1 b) f(x) = (x ) 4 + 3 c) f(x) = x 3 x d) f(x) = x 3 + x Int. Algebra Chapter 7 Summary page 9 of 14
e) f(x) = -x 3 + 3x f) f(x) = x 4 9x g) f(x) = -x(x + 1) (x ) 3 Int. Algebra Chapter 7 Summary page 10 of 14
SOLVING POLYNOMIAL EQUATIONS 1. Solve each equation algebraically. Find all real & complex roots. a) x 3 7x + 3x = 0 b) x 4 13x + 36 = 0 c) x /3 6x 1/3 + 5 = 0 d) d) x 3 + 8 = 0 Int. Algebra Chapter 7 Summary page 11 of 14
) Solve 3x x 4 = 0 by factoring. Explain how the coefficients of the factors relate to the coefficients in the original equations. 3) Solve: 3x 3 10x + 10x 4 = 0 4) Solve x 4 + x 3 x = 0 Int. Algebra Chapter 7 Summary page 1 of 14
5) Write a polynomial function of least degree with integral coefficients whose zeros include 3 and i. 6. The width of a rectangular prism is w centimeters. The height is cm less than the width. The length is 4 cm more than the width. If the volume of the prism is 8 times the measure of the length, find the dimensions of the prism. 7) An open-top box is to be made by cutting congruent squares of side length x from the corners of a 14 by 3- inch sheet of cardboard and bending up the sides. a) What is the volume of the box when x = 3? b) What is x when the volume is 500 cubic inches? c) For what x value(s) is the volume a maximum? Int. Algebra Chapter 7 Summary page 13 of 14
8) Think of a number. Subtract 7. Multiply by 3. Add 30. Divide by 3. Subtract the original number. The result is always 3. Use polynomials to illustrate this number trick. 9) The product of three consecutive integers is 1,30. Find the integers. 10) Find at least one zero for f(x) = x 3 + 4x 1. Estimate the zero accurate to at least one decimal digit. Do not use a graphing calculator. Int. Algebra Chapter 7 Summary page 14 of 14