MAT40S Mr. Morris Unit 2 Polynomial Expressions and Functions Note Package Lesson Homework 1: Long and Synthetic p. 7 #3 9, 12 13 Division 2: Remainder and Factor p. 20 #3 12, 15 Theorem 3: Graphing Polynomials p. 46 #3 13 4: Stupid Word Problems, I mean great word p. 61 #3 4, 5a, 5b, 6a reading this problems, stop Name:
Page 1 of 31 Lesson 1 Objective: Definition: Polynomial Function A function that has the form p x a x a x a x a x a Where p x is a function with variable x n is a whole number (1, 2, 3, etc) a to a are real number coefficients Starting Example: Determine which are polynomials, explain your reasoning
Page 2 of 31 Recall the process of long division: 327 12 The dividend is: The divisor is: Set up for long division: The quotient is: The remainder is: The solution can be written as
Page 3 of 31 Polynomial long division is a similar process to that of long division for numbers but follows a set pattern OR Example 1: x 2x 5 x 1
Page 4 of 31 Verify When long dividing the polynomial must be written in descending order (highest power to lowest power) If any power is missing then you use a zero coefficient for that power as a placeholder
Page 5 of 31 Example 3: 4x x 2x 3 x 3
Page 6 of 31 There is a quicker way to do polynomial long division but can be tricky unless proper steps are followed Definition: Synthetic Division Uses only coefficients of the polynomial to divide Must be in descending powers of x Add a 0 coefficient for any power that is missing The steps to properly synthetically divide are 1) Write out coefficients of the dividend in descending powers adding 0 where a power is missing 2) To the left, place the opposite number of the divisor 3) Bring down the first coefficient 4) Multiply by the divisor and add to the next coefficient 5) Repeat until finished 6) Beginning with the first number, write it with the variable that is one degree less than the dividend. The last number is the remainder. If the remainder is zero, this means the polynomials divide into each other perfectly!
Page 7 of 31 Example 4: 2x x 3x 5 x 2
Page 8 of 31 2x 3x 15 4x x 3 End of Lesson 1 Practice sheets p. 7 #3 9, 12 13
Page 9 of 31 Lesson 2 Objective: Recall factoring techniques 6x 7x 3 2x 3x 14x 49x 36y
Page 10 of 31 Determine if these binomials are factors of the polynomials Remember that if you divide a polynomial by a factor the remainder should be zero Is x 1 a factor of 3x 2x 1 We can easily what the remainder of a polynomial will be by using the remainder theorem Definition: Remainder Theorem A binomial in the form of x a can be used to determine the remainder of a polynomial by subbing in the value of x a into the polynomial
Page 11 of 31 Sub in the value x a into the p x to get the remainder p a Example 1: Find the remainder a) x 8x 5x 2 x 2 b) x 3x 9x 12 x 4 Example 2: Determine the value of k when 2x kx 5 is divided by x 3 if the remainder is 7
Page 12 of 31 We can easily see if a binomial is a factor of a polynomial by using the factor theorem Definition: Factor Theorem A binomial in the form of x a can be used to determine if it is a factor of a polynomial by subbing in the value of x a into the polynomial If subbing in the value x a into the polynomial p x the remainder will be 0 Example 3: Which of the following binomials are factors of p x x 3x x 3 x 1 or x 3
Page 13 of 31 We can use synthetic division and the factor theorem to factor polynomials with a degree higher than 2! 1) List all possible factors for the constant term 2) Begin evaluation p x with the factors you ve found 3) Stop when you find one that equals 0 4) Divide! You should get a remainder of 0, if you don t, you have a mistake somewhere! 5) Factor completely afterwards Example 4: Factor fully a) p x x 2x x 2
b) p x 2x 5x 11x 4 Page 14 of 31
Page 15 of 31 Example 5: When p x is divided by x 3 it has a quotient of 2x x 6 and a remainder of 4. Determine p x Example 6: When 2x kx 3x 2 is divided by x 2 the remainder is 4. Determine the value of k
Page 16 of 31 End of Lesson 2 Practice sheets p. 20 #3 12, 15
Page 17 of 31 Lesson 3 Objective: Recall the form of polynomial functions p x a x a x a x a x a The degree of a polynomial is the number of the highest power of the variable in the equation The leading coefficient is the coefficient of the highest power There are several types of polynomials that we will be graphing i) Linear (degree 1) ii) Quadratic (degree 3) iii) Cubic (degree 3) iv) Quartic (degree 4) v) Quintic (degree 5)
Page 18 of 31 The graphs of polynomial functions are always smooth and continuous. There are no sharp corners and can always be drawn without lighting up the pencil from the graph. The end behavior of a polynomial function is determined by the degree a leading coefficient. Degrees can either be Odd (1, 3, 5, etc) Even (2, 4, etc) Leading coefficients can either be Positive Negative From there, you can determine where the graph will end and start, as well as which direction it will travel left to right in. We will start with odd and even polynomials
Odd Polynomial Functions Page 19 of 31
Even Polynomial Functions Page 20 of 31
Page 21 of 31 A polynomial can have a degree of 0 but that would make it a constant function and it would be drawn as a straight, horizontal line Definition: Local Max and Min Point Local maximum points occur when the graph changes from increasing to decreasing Local minimum points occur when the graph changes from decreasing to increase Both make a smooth bump in the graph A polynomial function of degree n can have at most n x intercepts and at most (n 1) local max or min points
Page 22 of 31 GRAPHING POLYNOMIALS The zeros of any polynomial function, f x, correspond to the x intercepts of the graph and the roots of the equation,f x 0. The multiplicity of each factor determines the behavior of the graph at the zero. When a zero has an even multiplicity, the graph touches the x axis at the related x intercept, but does not cross it. When a zero has an odd multiplicity, the graph flattens out and crosses the x axis at the related x intercept.
Page 23 of 31 Example 1: Given the following polynomials determine the following i. Degree ii. Leading coefficient iii. End behavior iv. Y intercept v. Zeroes vi. Multiplicity at zeroes a) f x x 1 x 1 x 2
Page 24 of 31 b) g x x 2 x 1 Example 2: Sketch the graph of the following polynomial functions a) g x x 2 x 1
b) y x x 4 2 x x 5 Page 25 of 31
c) p x 2x 6x 4 Page 26 of 31
Example 2: Determine the function from this graph Page 27 of 31
Page 28 of 31 Example 3: Given the following characteristic of the polynomial function, determine the equation of h x i. X intercepts at 3, 1, 4 ii. Degree of 3 iii. Y intercept at 3/2 End of Lesson 3 Practice sheets p. 46 #3 13
Page 29 of 31 Objective: Lesson 4 Modelling and solving problems with polynomial functions is a staple for higher order word problems Example 1: A piece of cardboard 30cm long and 25cm wide is used to make a box with no lid. Equal squares of side length x cm are cut from the corners and the sides are folded up. Write a polynomial function to represent the volume, V, of the box in terms of x and sketch a graph to represent the polynomial.
Page 30 of 31 Example 3: A container with the shape of a rectangular prism has a volume in ft represented by V x 7x 28x 20. What are the factors that represent possible dimensions, in terms of x of the container
Page 31 of 31 Example 4: The product of four consecutive integers is x 6x 11x 6x where x is one of the integers. Determine possible expressions for the other three integers. End of Lesson 4 Practice sheets p. 61 #3 4, 5a, 5b, 6a