Algebra II Po lynomials: Operations and Functions 2013-09-25 IMPORTANT TIP: Throughout this unit,...

Algebra II

Polynomials: Operations and Functions

www.njctl.org

2013-09-25

Table of Contents

Adding and Subtracting Polynomials

Dividing a Polynomial by a Monomial

Characteristics of Polynomial Functions

Analyzing Graphs and Tables of Polynomial Functions

Zeros and Roots of a Polynomial Function

click on the topic to go to that section

Multiplying a Polynomial by a Monomial

Multiplying Polynomials

Special Binomial Products

Dividing a Polynomial by a Polynomial

Properties of Exponents Review

Writing Polynomials from its Zeros

Properties of Exponents Review

Return toTable ofContents

Exponents

Goals and Objectives

Students will be able to simplify complex expressions containing exponents.

Exponents

Why do we need this?Exponents allow us to condense bigger

expressions into smaller ones. Combining all properties of powers together, we can easily take a complicated expression and

make it simpler.

Properties of Exponents

Exponents

Multiplying powers of the same base:

(x4y3)(x3y)

Can you write this expression in another way??

Exponents

(-3a3b2)(2a4b3)

Simplify:

(-4p2q4n)(3p3q3n)

Work out:

Exponents

xy3 x5y4

. (3x2y3)(2x3y)

1 Simplify:

A m4n3p2

B m5n4p3

C mnp9

D Solution not shown

(m4np)(mn3p2)

Exponents

2 Simplify:

A x4y5

B 7x3y5

C -12x3y4

Exponents

er(-3x3y)(4xy4)

3 Work out:

A 6p2q4

B 6p4q7

C 8p4q12

Exponents

er2p2q3

4 Simplify:

A 50m6q8

B 15m6q8

C 50m8q15

Exponents

er.5m2q3 10m4q5

5 Simplify:

A a4b11

B -36a5b11

C -36a4b30

(-6a4b5)(6ab6)

Exponents

Dividing numbers with the same base:

Exponents

Simplify:

Exponents

Try...

6 Divide:

D Solutions not shown

Exponents

7 Simplify:

Exponents

8 Work out:

Exponents

9 Divide:

Exponents

10 Simplify:

Exponents

erPower to a power:

Exponents

Simplify:

Exponents

11 Work out:

Exponents

12 Work out:

Exponents

13 Simplify:

Exponents

14 Simplify:

Exponents

15 Simplify:

Exponents

Negative and zero exponents:

Exponents

Why is this? Work out the following:

Exponents

Sometimes it is more appropriate to leave answers with positive exponents, and other times, it is better to leave answers without

fractions. You need to be able to translate expressions into either form.

Write with positive exponents: Write without a fraction:

Exponents

Simplify and write the answer in both forms.

Exponents

Simplify and write the answer in both forms.

Exponents

Simplify: Teach

Exponents

Write the answer with positive exponents.

16 Simplify and leave the answer with positive exponents:

Exponents

17 Simplify. The answer may be in either form.

Exponents

18 Write with positive exponents:

Exponents

19 Simplify and write with positive exponents:

Exponents

20 Simplify. Write the answer with positive exponents.

Exponents

21 Simplify. Write the answer without a fraction.

Exponents

CombinationsExponents

Usually, there are multiple rules needed to simplify problems with exponents. Try this one. Leave your answers with positive exponents.

Exponents

When fractions are to a negative power, a short cut is to flip the fraction and make the exponent positive.

Try...

Exponents

Two more examples. Leave your answers with positive exponents.

Exponents

23 Simplify. Answer can be in either form.

Exponents

25 Simplify and write without a fraction:

Exponents

26 Simplify. Answer may be in any form.

Exponents

27 Simplify. Answer may be in any form.

Exponents

28 Simplify the expression:

29 Simplify the expression:

Adding and Subtracting Polynomials

Vocabulary

A term is the product of a number and one or more variables to a non-negative exponent.

The degree of a polynomial is the highest exponent contained in the polynomial, when more than one variable the degree is found by adding the exponents of each variable Term

degree degree=3+1+2=6

Identify the degree of the polynomials:

What is the difference between a monomial and a polynomial?

A monomial is a product of a number and one or more variables raised to non-negative exponents. There is only one term in a monomial.

A polynomial is a sum or difference of two or more monomials where each monomial is called a term. More specifically, if two terms are added, this is called a BINOMIAL. And if three terms are added this is called a TRINOMIAL.

For example: 5x2 32m3n4 7 -3y 23a11b4

For example: 5x2 + 7m 32m + 4n3 - 3yz5 23a11 + b4

Standard Form

The standard form of an polynomial is to put the terms in order from highest degree (power) to the lowest degree.

Example: is in standard form.

Rearrange the following terms into standard form:

Monomials with the same variables and the same power are like terms.

 Like Terms Unlike Terms  4x and -12x  -3b and 3a

 x3y and 4x3y  6a2b and -2ab2

Review from Algebra I

Combine these like terms using the indicated operation.

30 Simplify

31 Simplify

32 Simplify

To add or subtract polynomials, simply distribute the + or - sign to each term in parentheses, and then combine the like terms from each polynomial.

Example:

(2a2 +3a -9) + (a2 -6a +3)

Example:

(6b4 -2b) - (6x4 +3b2 -10b)

33 Add

34 Add

35 Subtract

36 Add

37 Add

38 Simplify

39 Simplify

40 Simplify

41 Simplify

42 Simplify

43 What is the perimeter of the following figure? (answers are in units)

Multiplying a Polynomialby a Monomial

Find the total area of the rectangles.

square units

To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication.Example: Simplify.

 -2x(5x2 - 6x + 8)

 (-2x)(5x2) + (-2x)(-6x) + (-2x)(8)

 -10x3 + 12x2 + -16x

 -10x3 + 12x2 - 16x

YOU TRY THIS ONE! Remember...To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication.

 Multiply: -3x2(-2x2 + 3x - 12)

 6x4 - 9x2 + 36xclick to reveal

More Practice! Multiply to simplify.

44 What is the area of the rectangle shown?

48 Find the area of a triangle (A=1/2bh) with a base of 5y and a height of 2y+2. All answers are in square units.

Multiplying Polynomials

sq.units

Area of the big rectangleArea of the horizontal rectanglesArea of each box

To multiply a polynomial by a polynomial, you multiply each term of the first polynomials by each term of the second. Then, add like terms.

Some find it helpful to draw arches connecting the terms, others find it easier to organize their work using an area model. Each method is shown below. Note: The size of your area model is determined by how many terms are in each polynomial.

6x2 4xy

12xy 8y2

Example: 

Example 2: Use either method to multiply the following polynomials.

The FOIL Method can be used to remember how multiply two binomials. To multiply two binomials, find the sum of ....

First terms Outer terms  Inner Terms  Last Terms

Example:

   First Outer Inner Last  

Try it! Find each product.

More Practice! Find each product.

49 What is the total area of the rectangles shown?

54 Find the area of a square with a side of

55 What is the area of the rectangle (in square units)?

How would you find the area of the shaded region?

56 What is the area of the shaded region (in square units)?

57 What is the area of the shaded region (in square units)?

Special Binomial Products

Square of a Sum

 (a + b)2  (a + b)(a + b)   a2 + 2ab + b2

The square of a + b is the square of a plus twice the product of a and b plus the square of b.

Example: 

Square of a Difference

 (a - b)2  (a - b)(a - b)   a2 - 2ab + b2

The square of a - b is the square of a minus twice the product of a and b plus the square of b.

Example: 

Product of a Sum and a Difference

 (a + b)(a - b) a2 + -ab + ab + -b2  Notice the -ab and ab    a2 - b2 equals 0.

The product of a + b and a - b is the square of a minus the square of b. 

Example:  outer terms equals 0.

Try It!  Find each product.

1.   

2.  

3.  

60 What is the area of a square with sides ?

Problem is from:

Click for link for commentary and solution.

A-APRTrina's Triangles

Dividing a Polynomial by a Monomial

To divide a polynomial by a monomial, make each term of the polynomial into the numerator of a separate fraction with the

monomial as the denominator.

Examples Click to Reveal Answer

62 Simplify

63 Simplify

64 Simplify

65 Simplify

Dividing a Polynomial by a Polynomial

Long Division of Polynomials

To divide a polynomial by 2 or more terms, long division can be used.

Recall long division of numbers.

MultiplySubtractBring downRepeatWrite Remainder over divisor

Long Division of Polynomials

To divide a polynomial by 2 or more terms, long division can be used.

MultiplySubtractBring downRepeatWrite Remainder over divisor

-2x2+-6x -10x +3 -10x -30 33

Examples

Example

Example: In this example there are "missing terms".   Fill in those terms with zero coefficients before dividing.

Examples

66 Divide the polynomial.

Characteristics ofPolynomial Functions

Polynomial Functions:Connecting Equations and Graphs

Relate the equation of a polynomial function to its graph.

A polynomial that has an even number for its highest degree is even-degree polynomial.

A polynomial that has an odd number for its highest degree is odd-degree polynomial.

Even-Degree Polynomials Odd-Degree Polynomials

Observations about end behavior?

Even-Degree Polynomials

Positive Lead Coefficient Negative Lead Coefficient

Odd-Degree Polynomials

Positive Lead Coefficient Negative Lead Coefficient

End Behavior of a Polynomial

Lead coefficient is positive

Left End Right End

Lead coefficientis negative

Left End Right End

Even- Degree Polynomial

Odd- Degree Polynomial

72 Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

A odd and positive

B odd and negative

C even and positive

D even and negative

A odd and positive

B odd and negative

C even and positive

D even and negative

A odd and positive

B odd and negative

C even and positive

D even and negative

A odd and positive

B odd and negative

C even and positive

D even and negative

Odd-functions not only have the highest exponent that is odd,but all of the exponents are odd.

An even-function has only even exponents.Note: a constant has an even degree ( 7 = 7x0)

Examples:

Odd-function Even-function Neither

f(x)=3x5 -4x3 +2x

h(x)=6x4 -2x2 +3

g(x)= 3x2 +4x -4

y=5x y=x2 y=6x -2

g(x)=7x7 +2x3

f(x)=3x10 -7x2

r(x)= 3x5 +4x3 -2

76 Is the following an odd-function, an even-function, or neither?

B Even

C Neither

B Even

C Neither

B Even

C Neither

B Even

C Neither

B Even

C Neither

An odd-function has rotational symmetry about the origin.

Definition of an Odd Function

An even-function is symmetric about the y-axis

Definition of an Even Function

81 Pick all that apply to describe the graph.

A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function

E Positive Lead Coefficient

F Negative Lead Coefficient

A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function

A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function

A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function

A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function

Zeros of a Polynomial

Zeros are the points at which the polynomial intersects the x-axis.

An even-degree polynomial with degree n, can have 0 to n zeros.

An odd-degree polynomial with degree n,will have 1 to n zeros

86 How many zeros does the polynomial appear to have?

Analyzing Graphs and Tables of Polynomial Functions

A polynomial function can graphed by creating a table, graphing the points, and then connecting the points with a smooth curve.

How many zeros does this function appear to have?

There is a zero at x = -1, a second between x = 1 and x = 2, and a third between x = 3 and x = 4. Can we recognize zeros given only a table?

Intermediate Value Theorem

Given a continuous function f(x), every value between f(a) and f(b) exists.

Let a = 2 and b = 4,then f(a)= -2 and f(b)= 4.

For every x value between 2 and 4, there exists a y-value between -2 and 4.

The Intermediate Value Theorem justifies saying that there is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

92 How many zeros of the continuous polynomial given can be found using the table?

-3 -12

93 Where is the least value of x at which a zero occurs on this continuous function? Between which two values of x?

-3 -12

95 What is the least value of x at which a zero occurs on this continuous function?

97 What is the least value of x at which a zero occurs on this continuous function? Give the consecutive integers.

Relative Maximums and Relative MinimumsRelative maximums occur at the top of a local "hill".Relative minimums occur at the bottom of a local "valley".

There are 2 relative maximum points at x = -1 and the other at x = 1 The relative maximum value is -1 (the y-coordinate).

There is a relative minimum at x =0 and the value of -2

How do we recognize "hills" and "valleys" or the relative maximums and minimums from a table?

In the table x goes from -3 to 1, y is decreasing. As x goes from 1 to 3, y increases. And as x goes from 3 to 4, y decreases.

Can you find a connection between y changing "directions" and the max/min?

When y switches from increasing to decreasing there is a maximum. About what value of x is there a relative max?

Relative Max:

click to reveal

When y switches from decreasing to increasing there is a minimum. About what value of x is there a relative min?

Relative Min:

click to reveal

Since this is a closed interval, the end points are also a relative max/min. Are the points around the endpoint higher or lower?

Relative Min:

Relative Max:

click to reveal

98 At about what x-values does a relative minimum occur?

99 At about what x-values does a relative maximum occur?

-3 -12

Finding Zeros of a Polynomial Function

Vocabulary

A zero of a function occurs when f(x)=0

An imaginary zero occurs when the solution to f(x)=0, contains complex numbers.

The number of the zeros of a polynomial, both real and imaginary, is equal to the degree of the polynomial.

This is the graph of a polynomial with degree 4. It has four unique zeros: -2.25, -.75, .75, 2.25

Since there are 4 real zerosthere are no imaginary zeros4 - 4= 0

When a vertex is on the x-axis, that zero counts as two zeros.

This is also a polynomial of degree 4. It has two unique real zeros: -1.75 and 1.75. These two zeros are said to have a Multiplicity of two.

Real Zeros -1.75 1.75

There are 4 real zeros, therefore, no imaginary zeros for this function.

106 How many real zeros does the polynomial graphed have?

107 Do any of the zeros have a multiplicity of 2?

108 How many imaginary zeros does this 8th degree polynomial have?

109 How many real zeros does the polynomial graphed have?

111 How many imaginary zeros does the polynomial graphed have?

112 How many real zeros does this 5th-degree polynomial have?

114 How many imaginary zeros does this 5th-degree polynomial have?

115 How many real zeros does the 6th degree polynomial have?

117 How many imaginary zeros does the 6th degree polynomial have?

Recall the Zero Product Property.

If ab = 0, then a = 0 or b = 0.

Find the zeros, showing the multiplicities, of the following polynomial.

or or or

There are four real roots: -3, 2, 5, 6.5 all with multiplicity of 1.There are no imaginary roots.

Finding the Zeros without a graph:

or or or or

This polynomial has five distinct real zeros: -6, -4, -2, 2, and 3.-4 and 3 each have a multiplicity of 2 (their factors are being squared)There are 2 imaginary zeros: -3i and 3i. Each with multiplicity of 1.There are 9 zeros (count -4 and 3 twice) so this is a 9th degree polynomial.

118 How many distinct real zeros does the polynomial have?

Find the zeros, both real and imaginary, showing the multiplicities, of the following polynomial:

This polynomial has1 real root: 2and 2 imaginary roots:-1i and 1i. They are simple roots with multiplicities of 1.

click to reveal

119 How many distinct imaginary zeros does the polynomial have?

F 5 Pu

120 What is the multiplicity of x=1?

This polynomial has two distinct real zeros: 0, and 1.There are 3 zeros (count 1 twice) so this is a 3rd degree polynomial.1 has a multiplicity of 2 (their factors are being squared).0 has a multiplicity of 1.There are 0 imaginary zeros.

To find the zeros, you must first write the polynomial in factored form.

There are two distinct real zeros: , both with a multiplicity of 1.There are two imaginary zeros: , both with a multiplicity of 1.

This polynomial has 4 zeros.

130 How many possible zeros does the polynomial function have?

131 How many REAL zeros does the polynomial equation

132 What are the zeros of the polynomial function , with multiplicities?

A x = -2, mulitplicity of 1

B x = -2, multiplicity of 2

C x = 3, multiplicity of 1

D x = 3, multiplicity of 2

E x = 0 multiplicity of 1

F x = 0 multiplicity of 2

133 Find the zeros of the following polynomial equation, including multiplicities.

A x = 0, multiplicity of 1

B x = 3, multiplicity of 1

C x = 0, multiplicity of 2

D x = 3, multiplicity of 2

134 Find the zeros of the polynomial equation, including multiplicities

A x = 2, multiplicity 1

B x = 2, multiplicity 2

C x = -i, multiplicity 1

D x = i, multiplicity 1

E x = -i, multiplcity 2

F x = i, multiplicity 2

135 Find the zeros of the polynomial equation, including multiplicities

A 2, multiplicity of 1

B 2, multiplicity of 2

C -2, multiplicity of 1

D -2, multiplicity of 2

E , multiplicity of 1

F , multiplicity of 2

To find the zeros, you must first write the polynomial in factored form.

However, this polynomial cannot be factored using normal methods. What do you do when you are STUCK??

RATIONAL ZEROS THEOREM

Make list of POTENTIAL rational zeros and test it out.

Potential List:

Test out the potential zeros by using the Remainder Theorem.

Remainder Theorem For a polynomial p(x) and a possible zero a, (x-a) is a factor of p(x) if and only if p(a) = 0.

1 is a distinct zero, therefore (x -1) is a factor of the polynomial. Use POLYNOMIAL DIVISION to factor out.

Using the Remainder Theorem.

This polynomial has three distinct real zeros: -2, -1/3, and 1, each with a multiplicity of 1.There are 0 imaginary zeros.

Find the zeros using the Rational Zeros Theorem, showing the multiplicities, of the following polynomial.

Potential List:

-3 is a distinct zero, therefore (x+3) is a factor. Use POLYNOMIAL DIVISION to factor out.

Remainder Theorem

This polynomial has two distinct real zeros: -3, and -1.-3 has a multiplicity of 2 (their factors are being squared).-1 has a multiplicity of 1.There are 0 imaginary zeros.

136 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem

B x = 1, mulitplicity 2

C x = 1, multiplicity 3

D x = -3, multiplicity 1

E x = -3, multiplicity 2

F x = -3, multiplicity 3

A x = -2, multiplicity 1

B x = -2, multiplicity 2

C x = -2, multiplicity 3

E x = -1, multiplicity 2

138 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem.

A , multiplicity 1

B , multiplicity 1

C , multiplicity 1

D , multiplicity 1

E x = 1, multiplicity 1

E x = , multiplicity 1

F x = , multiplicity 1

G x = , multiplicity 1

H x = , multiplicity 1

A x = -1, mulitplicity 1

B x = -1, mulitplicity 2

C x = , multiplicity 1

D x = , multiplicity 1

A x = -1, multiplicity 1

D x = 1, multiplicity 2

G x = , multiplicity 1

H x = , multiplicity 2

Writing a Polynomial Function from its Given

Write the polynomial function of lowest degree using the given zeros, including any multiplicities.

x = -1, multiplicity of 1x = -2, multiplicity of 2x = 4, multiplicity of 1

or or or

Work backwards from the zeros to the original polynomial.

Write the zeros in factored form by placing them back on the other side of the equal sign.

142 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

x = -.5, multiplicity of 1x = 3, multiplicity of 1x = 2.5, multiplicity of 1

x = 1/3, multiplicity of 1x = -2, multiplicity of 1x = 2, multiplicity of 1

x = 0, multiplicity of 3x = -2, multiplicity of 2x = 2, multiplicity of 1x = 1, multiplicity of 1x = -1, multiplicity of 2

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

x = -2

x = -1

x = 1.5x = 3

x = -2

x = -1

x = 1.5

or or or

When the sum of the real zeros, including multiplicities, does not equal the degree, the other zeros are imaginary.

This is a polynomial of degree 6. It has 2 real zeros and 4 imaginary zeros.

Real Zeros -2 2

A even and positive

B even and negative

C odd and positive

D odd and negative

A odd and positive

B odd and negative

C even and positive

D even and negative

A odd and positive

B odd and negative

C even and positive

D even and negative

Algebra II Po lynomials: Operations and Functions 2013-09-25 IMPORTANT TIP: Throughout this unit,...

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