Characteristics of Polynomial Functionsmathlau.weebly.com/.../9/14997226/chapter_3_polynomials.pdf · 2018-10-15 · Characteristics of Polynomial Functions: ... quotient polynomial

Chapter 3 – Polynomial Functions ● 1

Pre-Calculus 12

Characteristics of Polynomial Functions

Definition

A function of the form 01

12

22

21

1 ...)( axaxaxaxaxaxf nn

nn

nn ++++++= −

−−

− ,

where n is a ___________________________, x is a _______________ and 01221 ,,,...,,, aaaaaa nnn −− are

_____________ ______________. na is called the leading coefficient, and 0a is the constant term.

Example 1: Which functions are polynomials? Justify your answer. If they are polynomials, state the

degree, the leading coefficient, and the constant term of each polynomial functions.

Polynomial Polynomial?

Yes/No Degree

Leading

coefficient Constant term

5)( += xxf _______ _______ _______ _________

43)( xxg = _______ _______ _______ _________

xy = _______ _______ _______ _________

10597.023 234 +−+−= xxxxy _______ _______ _______ _________

5

19.0

3

2)( 2 +−= xxxf _______ _______ _______ _________

xxh

1)( = _______ _______ _______ _________

7)( 2

1

−= xxg _______ _______ _______ _________

344 4 +−−= xxy _______ _______ _______ _________

Characteristics of Polynomial Functions: Polynomial functions and their graphs can be analysed by

identifying the _________________, ____________________, _____________________________, and the

_____________________________________.

Definition: end behaviour – the behaviour of the __________________ of the function as x-values become

___________________________________

2 ● Chapter 3 – Polynomial Functions

Pre-Calculus 12

Example 2: Identify the following characteristics of each polynomial functions:

• degree and whether it is an even or odd

• the end behaviour of the graph of the function

• domain and range

• maximum or minimum value

• number of x-intercepts

• the value of the y-intercept

Constant Function

2)( =xf

Linear Function

12)( += xxf

Quadratic Function

12)( 2 −−= xxxf

Cubic Function

2332)( 23 −−+= xxxxf

Quartic Function

23432)( 234 +−−+= xxxxxf

Quintic Function

1241553)( 2345 ++−−+= xxxxxxf

Degree

End behaviour

Domain

Range

Max/Min

# of x-intercept(s)

Value of y-intercept

Degree

End behaviour

Domain

Range

Max/Min

# of x-intercept(s)

Value of y-intercept


Pre-Calculus 12

The graph of a Polynomial Function (Generalization)

Odd Degree

Positive leading coefficient Negative leading coefficient

Even Degree

Positive leading coefficient Negative leading coefficient


Pre-Calculus 12

Example 3: Given the graph of a polynomial, determine

• whether the graph represents an odd or an even-degree polynomial function

• determine whether the leading coefficient of the corresponding function is positive or negative

• state the number of x-intercepts

• state the domain and range

a)

b)

Example 4: Use the degree and the sign of the leading coefficient of each function to determine the end

behaviour of the corresponding graph. State the possible number of x-intercepts and the value of the y-

intercepts.

a) 163)( 24 −−= xxxf b) xxxxg ++−= 23 36)(

Assignment page 114 #1-4


Pre-Calculus 12

Recall: Long Division

652345

A similar can also be used to divide a polynomial, ie !453 23 +−+ xxx by a binomial, 3+x .

Long Division

Example 1: Divide 123 32 −+ xx by 2−x . Determine the quotient and reminder.

You Try: Divide 23 3252 xxx +−+ by 1−x . Determine the quotient and reminder.


Pre-Calculus 12

Division Statement for Division by x – a

)()(

)(

)(

ax

RxQ

ax

xP

−+=

−, 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) )1()142( 23 +÷−− xxx

b) )2()52( 3 −÷+− xxx

RESTRICTION!

Assignment page 124 #2, 5

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 Bthe

remainder when we divide a polynomial by a binomial.


Pre-Calculus 12

Example 3: Determine the remainder when the following polynomials are divided by the given binomial.

Also evaluate the given polynomial at the indicated value.

a) )1()35( 2 +÷+− xxx 35)( 2 +−= xxxp , evaluate )1(−p

b) ( ) ( )2732 23 −÷+−+ xxxx 732)( 23 +−+= xxxxp , evaluate )2(p

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 ax − the remainder is ______.

This changes the problem of finding the remainder from being a division question to being a substitution

question.

Example 4: Determine the remainder when the following polynomials are divided by the given binomial

a. )2()57( 23 +÷+−+ xxxx b. )2()35( 10 −÷−+ xxx

Example 5: When 10323 −−+ xkxx is divided by x + 2 the remainder is 20. Determine the value of k.


Pre-Calculus 12

Example 6: The remainder when 62 23 +−+ kxxx is divided by 3−x is –12. What is the remainder when the

polynomial is divided by x + 2.

Assignment page 124-125 #6(ace),7, 8, 9, 10, 14*

Factor Theorem

Example 1:

a) Determine the remainder when 2410)( 2 −−= xxxf is divided by i) )12( −x ii) )2( +x

b) Factor 2410)( 2 −−= xxxf

c) What do you notice?

The Factor Theorem

If ______________, then __________ is a factor of )(xp

OR

If a polynomial evaluated at a value k produces a value of 0, then kx − is a factor of the polynomial.

Example 2: Which of the following are factors of 306 23 −−+ xxx ?

a) 1+x b) 3+x c) 1−x d) 2−x


Pre-Calculus 12

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 kx − 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 3: Factor the following polynomial completely

a) 20118 23 −++ xxx b) 4214621 23 −++ xxx

Assignment page 133-134 #1, 2-6(ace), 7*


Pre-Calculus 12

Equations and Graphs of Polynomial Functions

Recall:

Example 1:

a) Solve 0)32()1( 2 =+− xxx

b) Graph the polynomial )2()2( 2 −+= xxxy using a table

of values.

x y

−5

−4

−3

−2

−1

0

1

2

3

4

c) What do you notice?

Definition: Multiplicity (of zero)

The number of ___________________________ of a polynomial function occurs. The shape of the graph

of a function close to a zero depends on its multiplicity.


Pre-Calculus 12

Example 2: For the following graph

a) Determine the zeros (x-intercepts) and their multiplicity

b) State the intervals where the function is positive and the intervals

where it is negative.

c) Write a possible equation for this graph.

Writing an Equation

If series of x-intercept points, an equation can be determined in standard form of

( )( )( ) ( )naxaxaxaxxf −−−−= ...)( 321 , 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

( )( )( ) ( )naxaxaxaxcxf −−−−= ...)( 321 , where c, a1, a2, a3…an are constants.

Example 3: Determine the cubic equation with zeros −3 (multiplicity of 2) and 1 and a y-intercept of 18.


Pre-Calculus 12

Graphing polynomials without a graphing calculator

Drawing a graph by hand it not as accurate, however you can the end behaviour, x-intercepts (zeros), y-

intercept and the location of the intervals were the function is positive and negative.

Example 4: Graph 43)( 23 +−= xxxf without a table of values

Assignment: page 147-150 #1-3(ac), 4, 7-8(abc), 9(ace),10, 14


Pre-Calculus 12

Modelling and Solving Problems with Polynomial Functions

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: Rectangular blocks of ice are cut up and used to build the front entrance of an ice castle. The

volume, in cubic feet, of each block is represented by 4875)( 23 −−+= xxxxv , where x is a positive real

number. What are the factors that represent possible dimensions, in terms of x, of the blocks?


Pre-Calculus 12

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.

x

x

12 cm

15 cm


Pre-Calculus 12

Example 3: Fran is 3 years older than Stan and Anne is 2 years younger than Stan. If the product of their

ages is 350. Determine how old each person is.

Assignment: worksheet

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Characteristics of Polynomial Functionsmathlau.weebly.com/.../9/14997226/chapter_3_polynomials.pdf · 2018-10-15 · Characteristics of Polynomial Functions: ... quotient polynomial