Unit 1: Polynomial Functions SuggestedTime:14 hours (Chapter 3 of the text) Prerequisite Skills Do the following: #1,3,4,5, 6a)c)d)f), 7a)b)c),8a)b), 9 Polynomial Functions A polynomial function is an algebraic expression consisting of at least 2 terms. Each consists of a real number and a variable(s) raised to some whole number exponent, which is called the of the term. A polynomial that contains 2 terms is called a. A polynomial that contains 3 terms is called a. Give an example of a: Binomial: Trinomial: The degree of the polynomial is determined by the highest found anywhere in the polynomial. 1 P a g e
Section 3.1: Characteristics of Polynomial Functions We have already studied two types of polynomial functions: and. We are now going to extend that study to include higher order polynomials such as functions. pg 107 Definition of a Polynomial Polynomial Function: a function of the form where n is a whole number x is a variable the coefficients of a n to a o are real number 2 P a g e
Key words: end behaviour, x-intercepts, degree, leading coefficient, # of turns (Bumps), y-intercept The following are polynomials: The following are NOT polynomials. Why? Example: Determine whether each of the following is a polynomial. Explain your reasoning. 3 P a g e
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The name of a polynomial is determined by its. The of a polynomial is determined by the of x found in the polynomial. The coefficient of the greatest power of x is called the. The term whose value is not affected by the variable is the. Note: The constant term is also the Examples: 5 P a g e
Characteristics of Polynomials Polynomial functions and their graphs can be analyzed by: Identifying the degree and sign of the leading coefficient End behavior Domain and range The number of x intercepts. Number of turns, aka bumps Activity Page 106 6 P a g e
SET A Graphs y x Set Function End Behaviour Degree Constant Term Leading Coefficient Number of x- Intercepts Number of turns (bumps) A Linear Quadratic Cubic Quartic Quintic 7 P a g e
SET B Graphs y x Set Function End Behaviour Degree Constant Term Leading Coefficient Number of x- Intercepts Number of turns (bumps) B Linear Quadratic Cubic Quartic Quintic 8 P a g e
y SET C Graphs x Set Function End Behaviour Degree Constant Term Leading Coefficient Number of x-intercepts Number of turns (bumps) C Linear Quadratic Cubic Quartic Quintic 9 P a g e
y Set D Graphs x Set Function End Behaviour Degree Constant Term Leading Coefficient Number of x-intercepts Number of turns (bumps) D Quadratic Cubic Quartic Quintic 10 P a g e
Basic features of the graphs of polynomial functions: The graph of a polynomial function is. The graph of a polynomial function has only. A function of degree n has at most. If the leading coefficient of the polynomial function is, then the graph rises to the. If the leading coefficient is, then the graph falls to the. The constant term is the of the graph. The number of x intercepts is between for degree for degree Odd vs. Even Degree Polynomials The polynomial functions that have the simplest graphs are the monomial functions. Even Degree (n is an even number) End behaviour is the for both ends 11 P a g e
Odd Degree (n is an odd number) End behaviour is for both ends Identify the features of the graph related to the function 12 P a g e
Matching a function with its graph Identify the following characteristics of the graph of each polynomial function: the type of function and whether it is of even or odd degree the end behaviour of the graph of the function the number of possible x intercepts whether the graph will have a maximum or minimum value the y-intercept 13 P a g e
Match the previous functions with their graphs below. 14 P a g e
Do your turn page 111 And Page 114-5 #1-6 15 P a g e
Section 3.2 You were introduced to factoring techniques for quadratic functions in Mathematics 1201. o You to express quadratics in factored form. In Mathematics 2200, these factoring techniques were extended to factor expressions with rational coefficients. So, by now you should be able factor polynomials (quadratics) of the form: To factor and solve polynomials of higher degrees some of the same techniques are used: First step is to look for : A) x 3 4x 2 + 3x 16 P a g e
B) 4 3 8 x (2x 1) 10 x (2x 1) 2x 1 Look for Difference of Squares A) x 2 9 B) x 2-5 C) 5x 4-80 D) 3 48 x ( x 1) 75 x( x 1) 17 P a g e
Higher degree polynomials with a quadratic form 2 ( ) a f x b f x c Example: Factor: A) x 4 2x 2-3 B) x 4 5x 2 +4 C) 4x 4 37x 2 +9 18 P a g e
Other Higher Degree Polynomials A) x 3 + 3x 2 4x 12 B) x 3 + 3x 2 x 3 C) D) x 3-2x 2 16x +32 = 0 Bonus! Factor: (i) 8x 5 40x 4 + 32x 3 x 2 + 5x 4 (ii) x 3 + x 2 8x - 12 19 P a g e
If Grouping doesn t work We need to determine the factor(s) through a different process. Polynomials can be using the same long process used to divide real numbers. Do you remember how to do? In this lesson we will learn, for polynomials: 1. 2. Polynomial Long Division!! Suppose we wanted to find the factors of 753. If we made a guess that 7 was a factor we would have to do long division. (If we did not have a calculator!) Long Division of Numbers Divide 753 by 7 Polynomial Long Division Divide x 3 + x 2 8x 12 by x - 1 20 P a g e
What is the significance of the remainder in both divisions? Write the division statement for each quotient. The division statement can also be written as: Let s check to see if x + 2 is a factor of x 3 + x 2 8x - 12 What is the significance of the remainder in this case? 21 P a g e
Write the division statement. Factor x 3 + x 2 8x 12 completely Draw the Graph y x 22 P a g e
Divide Synthetic Division Synthetic Division is a short cut method of dividing a polynomial by a divisor of the form. Consider the last problem: Instead of writing We write 23 P a g e
Examples: Divide 1. by Note: The divisor must have the form, so we write x + 1 as and put to the left of the coefficient row. 2. by Note: 24 P a g e
Practice: Divide f(x)=x 4 + x 3-23x 2 + 3x + 90 by the following divisors and complete the table below A) x + 1 B) x +2 C) x -1 D) x -3 Compare the remainders to f(a). What do you notice? 25 P a g e
Remainder Theorem When a polynomial,, is divided by, the remainder is Proof: Recall the division statement: Example: 1. A) Use the remainder theorem to determine the remainder when P(x) = x 4 x 3 +3x + 4 is divided by x + 2 B) Check your answer using synthetic division 26 P a g e
2.Find the remainder for the following: 5 3 2 3x 4x 2x x 1 A) B) x 3 51 6x 2x 4 x 1 3. Determine the value of k if the remainder is 5 for x 5 kx 3 5x 2 3x 1 x 2 4. Page 125 #15 27 P a g e
Application If the volume of a box is 3 2 x x x 4 19 14 What are the possible dimensions of the box in terms of x if the height is x 1? Section 3.2 do #1, 2, 3a), b), 4a) c), 5a) c), 6, 8a), b), 9, and 11. 28 P a g e
Section 3.2 Extra Practice 1. Use long division to divide x 2 x 15 by x 4. a) Express the result in the form P( x) R Qx ( ) x a x a. b) Identify any restrictions on the variable. c) Write the corresponding statement that can be used to check the division. d) Verify your answer. 2. Divide the polynomial P(x) x 4 3x 3 2x 2 55x 11 by x 3. a) Express the result in the form b) Identify any restrictions on the variable. c) Verify your answer. P( x) R Qx ( ). x a x a 3. Determine each quotient using long division. a) (3x 2 13x 2) (x 4) b) 3 2 2x 10x 15x 20 x 5 c) (2w 4 3w 3 5w 2 2w 27) (w 3) 4. Determine each remainder using long division. a) (3w 3 5w 2 2w 27) (w 5) b) 3 2 2x 8x 5x 2 x 1 c) (3x 2 13x 2) (x 2) 29 P a g e
5.Determine each quotient using synthetic division. a) (4w 4 3w 3 7w 2 2w 1) (w 2) b) 4 3 2 x 2x 8x 5x 2 x 2 c) (5y 4 2y 2 y 4) (y 1) 6. Determine each remainder using synthetic division. a) (3x 2 16x 5) (x 5) b) (2x 4 3x 3 5x 2 6x 1) (x 3) c) (4x 3 5x 2 7) (x 2) 7. Use the remainder theorem to determine the remainder when each polynomial is divided by x 2. a) 4x 4 3x 3 2x 2 x 5 b) 7x 5 5x 4 23x 2 8 c) 8x 3 1 8. Determine the remainder resulting from each division. a) (3x 3 4x 2 6x 9) (x 1) b) (3x 2 8x 4) (x 2) c) (6x 3 5x 2 7x 9) (x 5) 9. For (2x 3 5x 2 k x 9) (x 3), determine the value of k if the remainder is 6. 10. When 4x 2 8x 20 is divided by x k, the remainder is 12. Determine the value(s) of k. 30 P a g e
Warm Up Determine the remainder when 10x 4 11x 3 8x 2 + 7x + 9 is divided by 2x 3. Section 3.3 The Factor Theorem In the last section we discovered how to divide polynomial but the discovery of a factor was a matter of trial and error. In this section we discovery a way to reduce the number of trials. The Factor Theorem The factor theorem, states that a polynomial, P(x), has a factor if and only if. To see why this is true recall the remainder theorem from the last section: When a polynomial,, is divided by, the remainder is. So this factor theorem tells us that the remainder is. Thus So is a factor of P(x). 31 P a g e
Examples 1. Which of the following is a factor of P(x) = x 4 6x 2 10x + 3? A) x 1 B) x + 2 C) x -3 2. Use the factor theorem to decide whether x -1 is a factor of 2x 10 -x 7-1. Common Errors Mistakes are often made when substituting values into the equation, especially when the substitution involves negative values. Example: What is the error in this calculation? P(x) = x 2 - x + 3 divided by x + 1 P(-1) = -1 2-1 + 3 P(-1) = -1-1 + 3 P(-1) = 2 32 P a g e
This will avoid many mistakes, especially the. Example: Example 3 If x + 2 is a factor of x 3 + 5x 2 + kx + 4, find the value of k. Section 3.3 The Integral Zero Theorem The integral zero theorem is used to relate the of a polynomial and the of the polynomial. 33 P a g e
Integral ( ) Zero Theorem: If is an zero of a polynomial then is a of the of the polynomial. And is a factor of the polynomial Thus, the factors of the indicate of the polynomial. Example: Solve: x 2 3x 4=0 Note: Both and are factors of the constant term of. Example For the polynomial f(x)= x 3 3x 2 4x +12, A) What are the possible integral zeros of f(x)?? B) Use the factor theorem to determine roots (and factors) of f(x). 34 P a g e
Note: This method is restricted to polynomial functions with only and can be if the constant term has. In general, The factor theorem is used in conjunction with synthetic division to factor a polynomial. The is used to determine possible factors. Verify of the factors using the factor theorem. is then applied, Example: Find the factors of: A) P(x) = x 3 2x 2 5x + 6 resulting in a polynomial to be factored further. List the possible factors of the constant term. Then What?? 35 P a g e
B) f(x) = 4x 3-12x 2 + 5x + 6 C) g(x) = 4x 3-3x 2 + 7 D) 3x 3 + 8x 2 + 9x + 10 36 P a g e
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Section 3.4 Zeros, Roots and X-Intercepts In Mathematics 2200 you were introduced to the zeros of a quadratic function, the roots of the quadratic equation, and the x-intercepts of the graph. It is important that you distinguish between the terms zeros, roots and x-intercepts, and use the correct terms in a given situation. You could be asked to find the of an 3x 3 10x 2 23x 10 Find the of the 3x 3 10x 2 23x 10, Or determine the of the of f(x) = 3x 3 10x 2 23x 10 In each case, we are identifying the of the polynomial and to arrive at the solution. What are the x-intercepts of f(x) = 3x 3 10x 2 23x 10? 38 P a g e
Sketch the graph of f(x) y x Note: The of a polynomial, f(x), indicates the number of x-intercepts for its graph. For each real x-intercept there is a and a for the polynomial function f(x). The x-intercepts are found by finding the of, or by looking at where the graph of y=f(x). 39 P a g e
Practice Given the equation and the graph below, identify the zeros, roots and x-intercepts and explain how they are determined Multiplicity of Zeroes If a polynomial has a factor (x a) that is repeated times, then x = a is a zero of multiplicity,. 40 P a g e
Note: For a zero of multiplicity (single or triple root), the of the function at that zero. If a function has a zero of multiplicity (double or quadruple root), the sign. Examine each for the multiplicity of Zeroes What is the function rule for each graph? y x y x 41 P a g e
Examples: For each graph, determine: A) The least possible degree and name of polynomial B) The sign of the leading coefficient C) The x-intercepts D) The intervals where the function is positive (f(x) >0) and negative (f(x)<0) E) The factors of the function with the least possible degree. F) The value of the leading coefficient G) The function rule. i) y x 42 P a g e
ii) y x iii) y x 43 P a g e
Example Sketch the graph of each polynomial function. A) y = (x -1)(x + 2)(x 3) Degree Leading Coefficient End Behaviour y-intercept Interval(s) where function is positive or negative (Number Line method) y x 44 P a g e
B) y = -(x+1) 3 (x 3) Degree Leading Coefficient End Behaviour y-intercept Interval(s) where function is positive or negative y x 45 P a g e
C) y = (2x -1) 2 (x + 2) 2 (x -4) Degree Leading Coefficient End Behaviour y-intercept Interval(s) where function is positive or negative y x 46 P a g e
D) y =-2x 3 + 6x - 4 Degree Leading Coefficient End Behaviour y-intercept Interval(s) where function is positive or negative y x 47 P a g e
Word Problems 1. Which three consecutive integers have a product of -720? 2. Which four consecutive integers have a product of 3024? 48 P a g e
3. An open box is to be made from a 10 cm by 12 cm piece of tin by cutting x cm squares from each corner and folding up the sides. If the volume is to be 72 cm 3, what are the dimensions of the box? 4. Determine the equation with the least degree for a quartic polynomial function with zeros -2 (multiplicity 2) and 3 (multiplicity 2) and a constant term of -6. 49 P a g e