Grade 1 Pre-Calculus Mathematics Notebook Chapter 3 Polynomial Functions Outcomes: R11 & R1
3.1 Characteristics of Polynomial Functions R1 (p.106-113) Polynomial Function = a function of the form where f ( x) a x a x a x... a x a x a n n 1 n n n 1 n 1 0 where n = a whole number x = a variable a n and a 0 are real numbers Examples: f ( x) 3x 5 g x x x ( ) 3 17 h x x x x 3 ( ) 5 3 y x 7x 1 Vocabulary end behaviour degree constant term ( a 0 ) leading coefficient Page
Ex1: Identify the functions that are not polynomials and state why. For each, state the degree, leading coefficient, and the constant term of each polynomial function. a) b) c) d) Your turn: a) b) c) d) y = Page 3
Characteristics of Polynomial Functions Compare the graphs of even and odd degree functions. How does the leading term affect the general behaviour of the graph? Look at the graphs on page 109 of your text and list any generalizations or patterns you notice. Page 4
Characteristics of Polynomial Functions Match the following polynomials with its corresponding graph. 1. f(x) = x 3 4x + x +. g(x) = x 4 + 10x + 5x - 6 3. h(x) = x 5 + 5x 3 x + 1 4. p(x) = x 4 5x 3 + 16 a) b) y 40 0-3 - - 1 1 3 4 5 6 x y 6 5 4 3 1-0 - 40-60 - 4-3 - - 1-1 1 3 4 - - 3-4 - 5-6 x c) d) y y 3 4 16 8-4 - 3 - - 1 1 3 4-8 - 16 x 6 5 4 3 1-4 - 3 - - 1-1 1 3 4 - - 3-4 - 5-6 x Homework: Page 114 #1,, 3, 4, Page 5
3. The Remainder Theorem R11 (p.118-13) Long Division x 8x 15 Divide the following expression: = x 3 We can divide the expression above by using long division: x 3 x 8x 15 After you divide, your answer can be written in two forms: 1) Dividend Divisor remainder Quotient OR Divisor ) Dividend Divisor ( Quotient ) remainder Answer: Note: The restriction on the variable is x 3 since division by 0 is not defined. Note: To verify, multiply the divisor by the quotient and add the remainder. Page 6
Synthetic division Synthetic division is an alternate form of long division that we can use to divide polynomials. This type of division uses only the coefficients of each equation. Steps 1. Rearrange the equation in descending order.. Use the divisor to solve for x. Example: (x + ) x = 3. Write only the coefficients of the equation. If any are missing, fill in their spot with a zero. Make an L shape with the value of x outside. 4. Bring down the first coefficient. 5. Multiply by the divisor. 6. Add that number to the second coefficient. 7. Repeat steps 4-6 until there are no more coefficients to bring down. 8. Write the resulting numbers as the coefficients of a new equation. The last number will be the remainder. Divide: x 8x 15 x 3 Page 7
3 Ex1: Divide: x 5x 9 x 3 Answer: Ex: Divide: x 3 6x + 7 by x 4. Answer: Page 8
Remainder Theorem The remainder theorem allows us to obtain the value of the remainder without actually dividing. 3 Ex1: Determine the remainder when the polynomial P ( x) x 5x 17 x 1 is divided by the following binomials: a) x + 1 b) x 3 Synthetic division: Remainder Theorem: Homework: Page 14 #1,, 3 (choose ), 4 (choose 3), 5 (choose ), 6 (choose 3), 8 (choose ), 9, 11, C Page 9
3.3 The Factor Theorem R11 (p.16-133) The factor theorem tells us whether or not the divisor is a factor of the dividend. If there is no remainder (i.e. the remainder = 0), then the divisor is a factor. The factor theorem states that ( x a) is a factor of Px ( ) if and only if Pa ( ) 0. 3 Ex1: Determine whether or not x is a factor of ( x) x 6x 4 f. Ex: Completely factor 3 P( x) x 7x 6. To do this, we must first find factors of Let s use the Remainder Theorem. 3 P( x) x 7x 6. There must be an easier way than randomly guessing infinitely many times Page 10
Integral Zero Theorem Expand the following expression: 3 x 1 x x 5 x 4x 3x 10 Note: The factors of the polynomial are x 1, x + and x 5. The zeroes of the polynomial are 1,, and 5. Note: When we multiply all of the factors, the constant is + 10 which means that only factors of 10 can be factors of the polynomial. This is known as the integral zero theorem. Ex1: a) Find the possible factors of the following polynomial: x x 3 3x 6x 8 f b) Completely factor the polynomial above. Page 11
Ex: Determine all of the possible zeroes of the following polynomial: 3 x x 3x 8x 1 f b) Factor the polynomial. Page 1
4 3 Ex3: Factor P ( x) x 6x x 54 x 7. Homework: / Page 133 #1, a, f, 3c, 3e, 4-6 (choose 3 each), 7, 11, C1, C Page 13
3.3 Division of a Polynomial by a Binomial R11 1. Divide the polynomial P(x) x 4 3x 3 x 55x 11 by x 3. Express your answer in the form.. Determine the quotient of the following polynomials divided by binomials. a) (4w 4 3w 3 7w w 1) (w ) b) c) (5y 4 y y 4) (y 1) d) (3x 16x 5) (x 5) e) (x 4 3x 3 5x 6x 1) (x 3) f) (4x 3 5x 7) (x ) 3. Using the Remainder Theorem, determine the remainder when each of the following polynomials is divided by (x ). State whether or not (x ) is a factor of each. a) 4x 4 3x 3 x x 5 b) 7x 5 5x 4 3x 8 c) 8x 3 1 4. Divide each of the following and state whether or not the binomial is a factor of the polynomial. a) (3x 3 4x 6x 9) (x 1) b) (3x 8x 4) (x ) c) (x 3 5x 7x 9) (x 5) 5. Determine if (x 1) is a factor of each polynomial. a) 4x 4 3x 3 x x 5 b) 7x 5 5x 4 3x 8 c) x 4 3x 3 5x 7x 1 6. Determine is each polynomial has a factor of (x ). a) 3x 3 x 10x 5 b) x 4 3x 3 5x + 36 c) 3x 3 1x d) Explain how we know that (x + ) is not a factor of the polynomial in (a) without having to calculate the remainder. 7. Factor each polynomial below. a) x 3 x 13x 10 b) x 4 7x 3 3x 63x 108 c) x 3 x 6x 4 d) x 4 6x 5 e) x 4 4x 3 7x 34x 4 f) x 5 3x 4 5x 3 15x 4x 1 8. Given ( x 3 5x k x 9) (x 3), determine the value of k if the remainder is 6. 9. When 4x - 8x - 0 is divided by x + k, the remainder is 1. Determine possible values for k. 10. Each polynomial has x 3 as a factor. Determine the value of k in each case. a) kx 3 10x x 3 b) 4x 4 3x 3 x kx 9 Page 14
Solutions 1.. a) 4w 3-5w + 3w - 4 b) x 3 + 4x - 5 c) 5y 3-5y + 7y 8 d) 3x 1 e) 3 179 45 x 9x x 60 f) 4x 13x 6 x 3 x 3. a) -5 b) -44 c) -65 x + is not a factor of any of the polynomials since the remainder is never equal to 0. 4. a) 3 3 7x 14 x 1 x, no b) 3x, yes c) 44 7 x 5 5. a) no, r = -1 b) no, r = 43 c) yes, r = 0 6. a) no, r = 17 b) yes, r = 0 c) no, r = - d) The constant is 5 and is not a factor of 5. Therefore, there will always be a remainder. (integral zero theorem) 7. a) (x 1)(x )(x 5) b) (x 3) (x 3)(x 4) c) (x 1)(x 4)(x 6) d) (x 1)(x 1)(x 5)(x 5) e) (x 4)(x )(x 1)(x 3) f) (x 3)(x )(x 1)(x 1)(x ) 8. 9. 4 and - 10. a) 3 b) -7 x, no Page 15
3. Long Division & Synthetic Division 1. Divide the following polynomials using long division. a) x 3 3x x 5 x 1 c) x x x x 3 3 7 36 b) x 3 5x x 6 x d) 3 4x 9x 1 x. State any restrictions on the polynomials above. 3. Verify your answers from Question 1 by divide the polynomials using synthetic division. 3 4. The volume of a rectangular box, in cm, can be modeled by the polynomial function 3 V ( x) 3x x 1x 4. Determine expressions for the width and the length of the box if the height is x. 3. The Remainder Theorem 5. Use the remainder theorem to determine each remainder. a) 6x x 15 x 1 b) 3 x 5x 13x x 4 6. For each dividend, determine the value of k if the remainder is. a) x 3 5x 4x k x 1 b) x 3 4x kx 10 x 3 c) x x x x 4 3 5 7. For what value of m will the polynomial divided by x? x 1 and 3 P( x) x 6x mx 4 have the same remainder when it is 3.3 The Factor Theorem 8. Determine which of the following binomials are factors of a) x 1 b) 3 P( x) x 4x x 6. x c) x 3 d) x 9. Determine the possible integral zeros of each polynomial. a) 3 P( x) x 5x x 4 b) 4 3 P( x) x 3x x 1 c) P( x) x 3x 17x 30 10. An artist creates a carving from a block of soapstone. The soapstone is in the shape of a rectangular 3 prism whose volume, in cubic feet, is represent by V ( x) 6x 5x x 8 where x is a positive real number. Determine the factors that represent possible dimensions of the block of soapstone, in terms of x. 11. Factor: a) 3 P( x) x 4x x 6 c) b) 3 P( x) 3x 5x 6x 8 d) P x x x x 3 ( ) 3 3 P x x x x x 4 3 ( ) 3 7 7 18 Page 16
3.4 Equations and Graphs of Polynomial Functions R1 (p.136-147) Ex1: a) Determine the zeroes of the following cubic function: 3 7 6 3 1 f x x x x x x b) Determine the y-intercept of the function. c) Summarize what we know about this function. degree leading coefficient end behaviour zeroes y-intercept intervals (sign diagram) d) Sketch the graph. Page 17
Ex: a) Determine the roots of the following function: P ( x) x 3 4x x 4 b) Determine the y-intercept of the function. c) Summarize what we know about this function. degree leading coefficient end behaviour zeroes y-intercept intervals (sign diagram) d) Sketch the graph. Page 18
3.4 Sketching Polynomial Functions R1 1. State the zeroes and the y-intercept of each of the polynomial fucntions below. a) y x 3 x 5 x 1 b) y x x x 5 x 1 c) y 3 x x 1 x 8. Determine the intervals where the following functions are positive and negative (i.e. sign diagrams). a) y x 3 x 3 x 5 b) y 1 x x 4 x 5 c) y x x 1 x x 3 3. Sketch the following function. Be sure to include all intercepts. a) y x 1 x 1 x 5 b) y x x 3 x 4 c) y 1 x x x x 3 4. Explain what happens to the graph when the leading coefficient of a polynomial function is negative. 5. Sketch the graphs of the following functions. Be sure to include all intercepts. a) y x 3 9x 3x 15 b) y x 3 4x 4x 16 c) y x 3 4 3 9x d) y x x 16 x x 15 4 5 4 3 e) f ( x) x 13x 36 f) y x 3x 3x 89 x 94 x 10 Page 19
Solutions 1. a) x = -3, 5 and -1 y = -15 b) x = ±, 5 and -1 y = -0 c) x = 3, -1 and 8 y = -4. a) positive on the interval (-5,-3) and (3, ) negative on the interval (-, -5) & (-3, 3) b) positive on the interval (-, -4) and (1, 5) negative on the interval (-4,1) and (5, ) c) positive on the interval (-, -) and (-1, 0) and (3, ) negative on the interval (-,-1) and (0, 3) 3. a) b) c) 4. There is a reflection over the x-axis. The behaviour of the graph when x ± changes direction. 5. a) b) c) d) e) f) Page 0
Multiplicity of a Zero If ( ) Px has factor ( x a) n times, we say that x a is a zero of multiplicity n. For example, x 1 is a zero of multiplicity 3 y ( x 1) ( x )( x 4) x is a zero of multiplicity 1 x 4 is a zero of multiplicity 3 Multiplicity represents the number of times a factor is repeated. Multiplicity 1 Multiplicity Multiplicity 3 Multiplicity 4 Note: The effect of an even multiplicity is a bounce on the x-axis.. The effect of an odd multiplicity is a flattened area which is often hard to see without the use of technology. Ex1: Examples of multiplicities: a) y x 1 b) y x 3 Page 1
Ex: Sketch the following graphs: a) y x 1 x b) y x x 3 degree leading coefficient end behaviour zeroes y-intercept degree leading coefficient end behaviour zeroes y-intercept intervals (sign diagram) intervals (sign diagram) Ex.3 The zeroes of a polynomial function are, 3, and 5. Write an equation to represent this function. Ex. 4 The zeroes of a polynomial function are 0 and 4 (with a multiplicity of ). Write an equation to represent this function. Homework: Page 147 #1, 3, 4, 7, 8, 10, C1, C, C3 Page
Sketching polynomial functions Sketch the following functions. Be sure to clearly indicate the x and y-intercepts for each. 1. g ( x) ( x 1)( x 4)( x ). y x( x 1)( x 5) 3. f ( x) ( x 6)( x )( x 3)( x 8) 4. y ( x 1)( x 3) Page 3
5. f ( x) x ( x 7) y ( x 1) ( x 4) 6. y ( x 1) ( x 4) 7. 3 4 3 y x x x 6 8. f ( x) x 3x x 3 Page 4
3.4 Solving and Graphing Polynomial Equations R1 1. Solve each equation. a) (x 5)(x )(x 3)(x 6) 0 b) 4 5 x 3 x 1 0 x c) (3x 1)(4 x)(x 7) 0 d) (x 4) 3 (x ) 0 e) x 5 5 x 0 f) x 4 4 x 3 0. g) For questions (a) to (f), describe the degree of the polynomial as well as the end behaviour.. Using the graph of the polynomial function to the right: a) the smallest possible degree of the function b) the sign of the leading coefficient c) the x-intercepts of the graph and the factors of the function d) the intervals over which the function is positive and the intervals over which the function is negative. 3 3. Determine the equation of the smallest degree that corresponds to each polynomial function below. a) a cubic function with zeroes at 3 (of multiplicity ) and 1, and whose y-intercept is 18 b) a quintic function with zeroes at - (of multiplicity 3) and 4 (of multiplicity ) and whose y-intercept is 64. c) a quartic function with zeroes at 1 (of multiplicity ) and 5 (of multiplicity ) and whose y-intercept is 10. d) Sketch the functions (a) to (c). 4. Sketch the following graph. State all intercepts. a) 3 y x x 1 b) y x 3x x 3 c) y x 1 x 3 4 1 d) y x 1 x 3 e) y 3x x 3 f) y 8 x 1 x 1 x 5. Sketch the graph of each of the functions below. State all intercepts. a) y x 3 4x 5x b) f (x) x 4 19x 6x 7 c) g (x) x 5 14x 4 69x 3 140x 100x Page 5
Solutions 1. a) x = -5, -, 3 & 6 b) x = 5, 3 & 4 1 c) x = 1 3 g) For (a), 4 th degree polynomial with end behavior up in QI and QII. For (b), 3 rd degree polynomial with end behavior down in QIII and up in QI. For (c), 3 rd degree polynomial with end behavior up in QII and down in QIV. For (d), 5 th degree polynomial with end behavior up in QII and down in QIV. For (e), 4 th degree polynomial with end behavior down in QIII and QIV. For (f), 5 th degree polynomial with end behavior down in QIII and up in QI., 4 & 7 d) x = 0, -4 & - e) x = 5 & 5 f) x = 4 & -3. a) 3 b) negative c) 4,, 3 ; (x 4), (x ), (x 3) d) Positive intervals: ( 3, ) and (, 4); negative intervals: ( 4, 3) and (, 3. a) y (x 3) (x 1) b) 1 y x 3 x 4 c) d) «a» «b» «c» 4. a) b) c) d) e) e) f) 5. a) b) c) Page 6
3.4 Applications of Polynomial Functions R1 Ex1: The volume of air flowing into the lungs during one breath can be represented by the polynomial function where V is the volume in litres and t is the time in seconds. This situation can be represented by the graph below. y 6 5 4 3 1-6 - 5-4 - 3 - - 1-1 1 3 4 5 6 - - 3-4 - 5-6 x What does the x-axis represent? What does the y-axis represent? Determine any restrictions on the variables. Using the graph above, answer the following questions: a) Determine the maximum volume of air inhaled into the lungs. At what time during the breath does this occur? b) How many seconds does it take for one complete breath? c) What percentage of the breath is spent inhaling? Page 7
Ex: A block of snow measures 3m by 4m by 5m. The block melts in such a way that each dimension decreases in size at the same rate. At the end of a warm, sunny day, the block has a volume of 4m 3. a) Draw the initial block of snow. 4 x 3 x 5 x b) Write a polynomial function to represent this situation. c) Determine algebraically the new dimensions of the block. 4 (5 x)(4 x)(3 x) 0 x 3 1x 0 ( x 1)( x 0 ( x 1)( x 47x 36 11x 36) 11x 36) Page 8
Ex3: The length of a swimming pool is 10m larger than the depth. The width of the pool is 3m larger than the depth. The City of Winnipeg charges $ per m 3 of water used to fill the pool. The bill to fill the above pool is $40. a) Represent this situation algebraically in terms of the pool s depth, d. b) Determine all possible values for the depth of this pool. 10 d( d 10)( d 3) 0 d 3 13d 0 ( d )( d 30d 10 15d 60) c) Determine the real dimensions of the pool. Page 9
Ex4: A box is assembled by cutting the corners of a piece of cardboard and then folding up the remaining sides. A piece of cardboard has a length of 30cm and a width of 0cm. A square with sides measuring x cm is cut from each of the corners of the cardboard as shown in the diagram below. a) Write an algebraic expression that represents the volume of this box. b) We would like a box with a volume of 1000cm 3. Determine the dimensions of the box that could be created with this piece of cardboard. 1000 0 4x 0 x x(30 x)(0 x) 3 3 100x 5x 0 ( x 5)( x 600x 1000 150x 50 0x 50) Homework: Page 150 #1, 15, 16, 18 Page 30
3.4 Applications of Polynomial Functions R1 1. Write an algebraic expression to calculate the product of 3 whole consecutive numbers if the smallest number is x and the product is the function P(x). Next, determine the 3 numbers if the product equals 504.. The product of 3 odd consecutive numbers is 315. Find the 3 numbers using a polynomial function. 3 3. The volume of a rectangular aquarium is modeled by the equation V ( x) x 19 x 110 x 00. The depth of the aquarium can be represented by the expression x 5. Write a polynomial to represent the length and width of the aquarium. 4. The Pan-Am pool contains 8000m 3 of water. It has dimensions x for the depth, x + 6 for the length and 5x for the width. a) Determine all possible values for x. b) Determine the dimensions of the pool. 5. A rectangular box has square ends. The measure of the box s length is 1 cm longer than the width. The volume is 135 cm 3. Determine the dimensions of the box. 6. A rectangular prism measures 10 cm by 10 cm by 5 cm. When each dimension is increased by the same quantity, the volume becomes 1008 cm 3. Determine the dimensions of this new rectangular prism. 7. An open box with locking tabs is to be made from a square piece of cardboard with side length 8 cm. This is done by cutting equal squares of side length x cm from the corners and folding along the dotted lines as shown.. a) Write a polynomial equation to represent the volume, V, of the box in terms of x. b) Sketch the polynomial function that represents this situation. c) Determine the approximate value of x that maximizes the volume of the box to the nearest centimetre. Note: use the graph to make your estimation. 8. A large ice cube with dimensions 6cm by 4cm by 4cm is placed on the counter where it uniformly melts. Upon returning hours later, the ice cube has melted to a sixth of its original volume. Determine algebraically the new dimensions of the ice cube. 9. The dimensions of a box are x 9 cm, 75 5 x cm, and 3x 13 cm. Determine the possible dimensions of the box if each dimension is a whole number and if the volume of the box is 8000cm 3. Page 31
Solutions 1. P x x x 1 x and 9, 8 and 7. 5, 7 and 9 3. width: x + 4, length: x + 10 4. a) x = -10, -6 and 10 b) x = 10 is the only real value of x. Thus, the dimensions are 10m by 16m by 50m 5. 3 cm by 3 cm by 15 cm 6. 1 cm by 1 cm by 7 cm 7. a) V( x) 8x(14 x)(7 x) b) c) x = 3cm 8. 4 cm by cm by cm 9. 10cm, 50cm and 16cm or 8cm, 100cm and 10cm Page 3