Sec 2.1 Operations with Polynomials Polynomial Classification and Operations

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2 Sec.1 Operations with Polynomials Polynomial Classification and Operations Name Examples Non-Examples Monomial 1. x 4 degree:4 or quartic 1. x 4 (one term). a degree: or quadratic. 5 m. 5 degree:0 or constant. t Binomial (two terms) Trinomial (three terms) Polynomial (one or more terms) 1. n n degree: or cubic. p degree:1 or linear(monic). a b 4 + a 4 b 5 degree:9 or nonic 1. x + x degree: or cubic. d(d + d 4 ) degree:5 or quintic 1. x 4 + x 5x + 1 degree:4 or quartic. 5y 6 degree:6 or sextic 1. x + x 6x 4 + 1x degree:4 or quartic 1. x+1 x. c 1. x + x 5. x + x 5 1. q + p q. x + x Name: 1. EXPAND and SIMIPLIFY a. (7x ) ( x) b. (5x x 4 x 9x ) + (x +x 5x 7) c. ( x 5) 8x d. y 5x 6y x 7 x e. x 5x 6 x x x g. x x 5 h. x 5 f. 5x 8 5x 9x 11x 5 M. Winking Unit -1 page 7

3 (1 Continued). EXPAND and SIMIPLIFY 4y y y j. - 6y (y - y - 7) i. k. x x 5 l. a a a 4 m. x x 4x n. o. Determine an expression that represents: Perimeter = Perimeter = Determine an expression that represents: Area = Area= M. Winking Unit -1 page 8

4 Sec. Operations with Polynomials Pascal s Triangle & The Binomial Theorem 1. Expand each of the following. Name: a. (a + b) 0 b. (a + b) 1 c. (a + b) d. (a + b) e. (a + b) 4 f. (a + b) 5. Create Pascal s triangle to the 7 th row. M. Winking Unit - page 9

5 . Using Pascal s Triangle expand (a b) 4 The Binomial Theorem permits you to determine any row of Pascal s Triangle Explicitly. The Binomial Theorem is shown below: (a + b) n = ( n C 0 )(a) n (b) 0 + ( n C 1 )(a) n 1 (b) 1 + ( n C )(a) n (b) + + ( n C n 1 )(a) 1 (b) n 1 + ( n C n )(a) 0 (b) n 4. Using the Binomial Theorem expand (x y) 6 M. Winking Unit - page 0

6 Use the Binomial Theorem to answer the following: 5. What is just the 4 th term of (c + 4d) 7 CC 6. What is just the 7 th term of (q p) 6 7. What is just the coefficient of the rd term of (t + 5m) 8 8. Which term of (5a b) 9 could be represented by ( 9 C 7 )(5a) ( b) 7 9. The Binomial Theorem also has some applications in counting. For example if you wanted to know the probability of 6 coins being flipped and the probability that 5 of the flipped coins will land on heads by expanding. First, expand (h + t) 6 using which ever method you would prefer. Each coefficient represents the number of different ways you can flip a the 6 coins that way. (e.g. 15h 4 t suggests there are 15 different ways the 6 coins could land with 4 heads up and tails up) a. Determine the probability of having 5 coins land heads up. b. Determine the probability of having or more tails landing heads up. M. Winking Unit - page 1

7 Sec. Operations with Polynomials Dividing Polynomials Name: 1. Divide each of the following polynomials by the suggested monomial. a. 5 a 4a 8a b. 6x 7x 48x 6x 5 1m 0m m c. 4m 5 4. (REVIEW) Complete the following long division problem: Use long division to divide the following polynomials. x 4 5x x 8x x x x x x x Use long division to divide by x 4 x 7x 10x 6x 5. Use long division to divide: 4 x x x 8x 6 x M. Winking Unit - page

8 6. Use long division to divide4x 4 x x by x 1 7. Use long division to divide: 5 4 x x x x x 5 6 x x 8. Use long division to find the quotient of and x x x 1 9. Use long division to determine. 6x 4 8x 11x 7x 6 x x 10. Rewrite x x 4x 5 as a nested polynomial. 11. Use the nested polynomial to easily evaluate x x 4x 5 when x =. 1. Use the following format to quickly evaluate x x 4x when x =. M. Winking Unit - page

9 1. Use long division to find the following quotient: x x 4x 5 x (Compare the answer from problem #1 with problem #11.) 14. Use synthetic division to divide 15. Use synthetic division to divide x 4 7x 17x 1x 6 x x 4 x x x 6 x 16. Use synthetic division to divide 4 x 5x 7x 6 x 17. Use synthetic division to evaluate x x 4x at x = 18. Use synthetic division find the remainder of x 4 10x 6x 5x 7 x Which of the following is a factor of 5 x x 6x 5x 6 a. (x+1) b. (x ) c. (x+) d. (x 1) M. Winking Unit - page 4

10 Sec.4 Operations with Polynomials Composition of Functions 1. Consider the following functions. Name: f(x) = 6x g(x) = x h(x) = x + p(x) = a. Determine (f + g)(x) b. Determine (f + h)(x) c. Determine (g f)(x) d. Determine (p g)(x) e. Determine (f g)() f. Determine ( f g ) (x). Given the following partial set of values of function evaluate the following. a. Determine f(1) g() b. Determine (f + g)(). Given the following partial set of values of function evaluate the following. a. Determine f(4) + g(1) M. Winking Unit -4 page 5

11 4. Consider the following functions. f(x) = 6x g(x) = x h(x) = x + p(x) = a. Determine (f h)(1) b. Determine (g f)() c. Determine (f g)(x) d. Determine (g h)(x) 5. Given the following partial set of values of function evaluate the following. a. Determine (f g)() b. Determine (g f)(0) 6. Given the following partial set of values of function evaluate the following. a. Determine (f g)(0) M. Winking Unit -4 page 6

12 x Given the length of a rectangle can be described by the function f(x) = 6x and the width of the same rectangle can be described by g(x) = x + 1 a. Determine (f g)(x) and tell what it would represent as far as the rectangle is concerned. 6x b. Determine an expression that represents the perimeter of the rectangle. 8. A pyramid is created from a square base where each side is 6 cm. The volume of the pyramid can be given by the function, V(h) = 1h. The function to calculate the height of the same square based pyramid given the slant height could be described by H(s) = s 9 where s is the slant height in cm. a. Evaluate V(H(5)) = b. Explain what V(H(5)) represents. 9. After being filled a party balloon s volume is dependent of the temperature in the room. The volume of the balloon can be modeled by V(c) = c where the volume, V, is measured in cubic centimeters and the temperature, c, is measured in degrees Celsius. A function that enables you to change from degrees Celsius (C ) to degrees Fahrenheit (F ) is C(f) = 5 (f ). 9 Show the composition of function required to determine the volume of the balloon when the temperature is 98 F. 10. A square is increased by doubling one side and decreasing the other sided by units. Let x be the length of a side of the square. Create a function that would represent a change in the area of the rectangle after the transformation. M. Winking Unit -4 page 7

13 Sec.5 Operations with Polynomials Inverses of Functions Inverse of a Function conceptually Name: f x 4 1 x f x 1. Find the inverse functions of the following. a. f x 5x b. g x x 1 5 x g x x 6 c. h x 6 d. M. Winking Unit -5 page 8

14 . Given the graph create an inverse graph and determine if the inverse is a function. a. Create an inverse of the graph shown Is the inverse a function? CIRCLE ONE: YES NO b. Create an inverse of the graph shown Is the inverse a function? CIRCLE ONE: YES NO c. Create an inverse of the graph shown Is the inverse a function? CIRCLE ONE: YES NO d. Create an inverse of the graph shown Is the inverse a function? CIRCLE ONE: YES NO M. Winking Unit -5 page 9

15 . Which two functions could be inverses of one another based on the partial set of values in the table? 4. Find the inverse functions of the following using the x y flip technique. x 1 5 a. g x b. h x x 1 1 ; x c. f x x ; x x d. m x x x 1 1 ; x M. Winking Unit -5 page 40

16 5. Find the inverse functions of the following using any method: a. f x x x b. g x x 4 ; x 0 6. Verify which of the following are inverses of one another by considering f g x and g f x f x 4x f x x 1 a. g x x 4 b. g x x 1 c. f x g x x x d. f x x g x 1 x 1 M. Winking Unit -5 page 41