Unit 3 – Polynomial Functions, Equations and Inequalities
Chapter 1 – Polynomial Expressions and Functions
Name: _______________
[pic]
| |Ch. 1.1 Dividing a Polynomial by a Binomial |Date: |
Recall: Long Division
[pic]
Long division can also be used to divide a polynomial by a binomial
Example 1: Divide[pic]by [pic]. Determine the quotient and reminder.
You Try: Divide[pic]by [pic]. Determine the quotient and reminder.
|Division Statement for Division by x – a |
|[pic], where P(x) is the original polynomial, (x – a) is the binomial divisor, Q(x) is the quotient polynomial (which has a degree 1 less than P(x)), and R is |
|the reminder (which is a constant). |
Synthetic division can also be used to divide a polynomial by a binomial instead of long division
Example 2: Use synthetic division to divide. Write a division statement.
a) [pic]
b) [pic]
Example 3: When [pic], where p is a constant, is divided by [pic], the remainder is –24. Determine p.
Assignment page 7-12 #3-9, 11, 12; page 13 #1-2
| |Ch. 1.2 Factoring Polynomials |Date: |
Reminder Theorem
When we are trying to factor a polynomial, we would like to be able to determine whether a certain binomial divides evenly into it. It would therefore be very helpful if we had a quick way of determining the remainder when we divide a polynomial by a binomial.
Example 1: Determine the remainder when the following polynomials are divided by the given binomial. Also evaluate the given polynomial at the indicated value.
|[pic] |[pic], evaluate [pic] |
|[pic] |[pic], evaluate [pic] |
What is special about the indicated value when evaluating the polynomial?
What do you notice about synthetic division and evaluating the polynomial at the given value?
|The Remainder Theorem |
| |
|When the polynomial p(x) is divided by [pic]the remainder is p(a). |
| |
This changes the problem of finding the remainder from being a division question to being a substitution question.
Example 2: Determine the remainder when the following polynomials are divided by the given binomial
a. [pic] b. [pic]
Example 3: When [pic]is divided by x + 2 the remainder is 20. Determine the value of k.
Example 4: The remainder when [pic]is divided by [pic]is –12. What is the remainder when the polynomial is divided by x + 2.
Factor Theorem
Recall: Factor [pic]
If the roots (solutions) are plugged back into the equation f(x), what will you get?
|The Factor Theorem |
|If ______________, then __________ is a factor of [pic] |
|OR |
|If a polynomial evaluated at a value k produces a value of 0, then [pic]is a factor of the polynomial. |
Example 5: Which of the following are factors of [pic]?
a) [pic] b) [pic] c) [pic] d) [pic]
What is the maximum numbers of factors that the polynomial above could have?
Once one factor is found, how can other factor(s) be found?
We now have an easy way of determining whether a binomial is a factor of a given polynomial. The question now is: “Which binomial should we test?”
|The Factor Property |
|If [pic]is a factor of a polynomial, then k must be a factor of the __________ ___________ of the polynomial. |
The factor property allows us the significantly narrow our search for possible factors.
Example 6: Factor the following polynomial completely
|[pic] |[pic] |
Assignment page 20-24 #4-12; page 26 #1-2
For Graphing Polynomial Functions Notes, see page 32 in Workbook. Assignment page 34-36 #1-2
| |Ch. 1.4 Relating Polynomial Functions and Equations |Date: |
|Definition |
|A function of the form [pic], |
|where n is a ___________________________, x is a _______________ and [pic]are _____________ ______________. [pic]is called the leading coefficient, and |
|[pic]is the constant term. |
Example 1: Which functions are polynomials? Justify your answer. State the degree, the leading coefficient, and the constant term of each polynomial functions.
|Polynomial |Polynomial? |Degree |Leading coefficient |Constant term |
| |Yes/No | | | |
|[pic] |_______ |_______ |_______ |_________ |
|[pic] |_______ |_______ |_______ |_________ |
|[pic] |_______ |_______ |_______ |_________ |
|[pic] |_______ |_______ |_______ |_________ |
|[pic] |_______ |_______ |_______ |_________ |
|[pic] |_______ |_______ |_______ |_________ |
|[pic] |_______ |_______ |_______ |_________ |
|[pic] |_______ |_______ |_______ |_________ |
Characteristics of Polynomial Functions: Polynomial functions and their graphs can be analysed by identifying the degree, end behaviour, domain and range, and the number of x-intercepts.
Definition: end behaviour – the behaviour of the y-values of the function as x-values become more positive and negative
Definition: local minimum/maximum point – a point where the graph changes from decreasing to increasing or increasing to decreasing
Example 2: Identify the following characteristics of each polynomial functions:
|Degree and whether it is an even or odd |number of local maxima or minima values |
|the end behaviour of the graph of the function |number of x-intercepts |
|domain and range |the value of the y-intercept |
Linear Functions
[pic] [pic]
Quadratic Functions
[pic] [pic]
Cubic Functions
[pic] [pic] [pic]
Quartic Functions
[pic] [pic] [pic]
Quintic Functions
[pic] [pic] [pic]
Example 3:
What might you expect the degree of the following function to be?
What does the degree of a polynomial tell you about the graph?
What does the leading coefficient tell you about the graph?
What does the constant term tell you about the graph?
Predict what you the graph of [pic]to probably look like.
The graph of a Polynomial Function
The graph of any polynomial function is a _______________, _____________________ curve. The domain of all polynomial functions are _____________. The constant term is the ___________________. The maximum number of peaks and valleys is ____________________________. The orientation of the graph is determined by the sign of the ________________ coefficient.
Odd Degree
Positive leading coefficient Negative leading coefficient
[pic] [pic]
Even Degree
Positive leading coefficient Negative leading coefficient
[pic] [pic]
Graphing Polynomial Functions
Example 4: Graph [pic]with a table of values.
|x |y |
|−5 | |
|−4 | |
|−3 | |
|−2 | |
|−1 | |
|0 | |
|1 | |
|2 | |
|3 | |
|4 | |
Example 5: Factor [pic]by using synthetic division or long division, then graph it.
For more accurate graph it is best if you factor the polynomial and use a table of values.
Definition: multiplicity (of zero) – the number of times a zero of a polynomial function occurs.
If a graph has a factor x – a repeat more than once, the zeros will have different effects on graph.
• If [pic]where the multiplicity is 1, then at the x = a the graph will pass through.
• If [pic]where the multiplicity is 2, then at the x = a the graph will just touch.
• If [pic]where the multiplicity is 3, then at the x = a the graph will slide through.
Example 6: Sketch the graph of[pic].
Writing an Equation
If series of x-intercept points, an equation can be determined in standard form of
[pic], where a1, a2, a3…an are constants.
In addition, if a non-zero point is provided, then a leading coefficient, c, can also be determine and form an equation of the form
[pic], where c, a1, a2, a3…an are constants.
Example 7: Write the equation of a quadratic function with zeros −5 and 12 which passes through (3, −5).
You Try: Write the equation of a cubic function with zeros 2, 3, and −4 and which passes through (−1, 8).
Example 8: Write the equation of a quartic function with zeros at −3 of multicity 2, zeros at 2 and 8 of multicity 1 and with[pic].
Assignment: page 46-52 #3-13; page 54-55 #1-2
| |Ch. 1.5 Modelling and Solving Problems with Polynomial Functions |Date: |
Many real-life situations can be modelled by polynomial functions. The solutions may be rational and can be solved by methods in this chapter. Other solutions are irrational, and can be only solved by graphing calculator. Here are some examples.
Example 1: A box constructed such that the length is twice the width and the height is 2 cm longer than the width, with a volume of 350 cm3. Find the dimensions of the box.
Example 2: An open rectangular box is constructed by cutting a square of length x from each corner of a 12 cm by 15 cm rectangular piece of cardboard, then folding up the sides. What is the length of the square that must be cut from each corner if the volume is 112 cm3.
Example 3: Lesley is 5 years younger than Clara. Mike is 2 years younger than Clara. Thomas is 3 years older than Clara. The product of their ages was 61136 greater than the sum of their ages. How old was Clara and each friend?
Assignment: page 61-66 #3-11; page 67-68 #1-2
Chapter Review Assignment: page 28-31 #1-7; page 72-77 #1-9; page 78-80 #1-7
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Degree:
End behaviour:
Domain:
Range:
Local Maxima/Minima:
x-intercept:
y-intercept:
Degree:
End behaviour:
Domain:
Range:
Local Maxima/Minima:
x-intercept:
y-intercept:
Degree:
End behaviour:
Domain:
Range:
Local Maxima/Minima:
x-intercept:
y-intercept:
Degree:
End behaviour:
Domain:
Range:
Local Maxima/Minima:
x-intercept:
y-intercept:
Degree:
End behaviour:
Domain:
Range:
Local Maxima/Minima:
x-intercept:
y-intercept:
x
x
12 cm
15 cm
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