math-3 lesson 4-4 roots and graphs of higher degree...

Math-3Lesson 4-4

Roots and Graphs of Higher Degree Polynomials

Quiz 4-4

)3()4532( 23 xxxx2. What is the quotient and the remainder for

the following:

1. What are the possible rational roots of the

following polynomial?

3523 23 xxxy3

1 ,3 ,1

3

8432 2

xxx

What you’ll learn about

• Three Major Theorems

• How multiplicities affect the graph of a polynomial

• Combining “end behavior”, real number zeroes and their multiplicities to draw a rough graph of a polynomial.

… and why

These topics provide the complete story about

the zeros and factors of polynomials with real number coefficients.

Complex Numbers

b

a

b

a

12

Rational #’s

Irrational #’s

Whole #’s 0, 1, 2, 3 …

Integers: …-2, -1, 0, 1, 2 …

Rational #’s ( ) (1/2, 2/3, 5/1, 0.5, 0.666…, 5)b

a

e2

0.1010010001…

1

i

Imaginary #’sReal #’s

Complex Number Defined

a + bi

A member of either (or both) the real

and imaginary number families.

Real part Imaginary part

3 + 2i

5, -6, -7i, 2 + 3i

Standard Form

Your turn: Simplify

(2 – 3i)(4 + i)

(2 – 7i)(2 + 7i)

Multiplying Complex Numbers

(2 + 3i) (5 – i) = ?

(x + 1)(2x – 3) = ?(background)

(2)(5) + (2)(-i) + (3i)(5) + (3i)(-i)

10 + (-3)(i)(i) + (15 – 2)i

13 + 13i

10 + 3 + 13i

Polynomial Review

325 34 xxxy

What is the degree of the polynomial?

What is the lead coefficient of the polynomial?

What is the end behavior of the polynomial?

Positive lead coefficient Odd degreeSame as: xy

Down on left, up on right

325 27 xxxy

Review the Fundamental Theorem of Algebra

A polynomial function of degree n has n complex

zeros (real and imaginary). Some of these zeros

may be repeated.

How many solutions (real or imaginary)

does this polynomial have?

Does this theorem tell you how to find the zeroes?

Are the complex zeroes of f(x).

The are not necessarily distinct numbers;

some may repeat.

Linear Factorization Theorem

))...()(()( 21 nzxzxzxaxf

If f(x) is a polynomial function (degree n > o),

then f(x) has precisely “n” linear factors and

where “a” is the leading coefficient of f(x) and

nzzz ,...,, 21

iz

Contrast the

Linear Factorization Theorem (LFT)

to the

Fundamental Theorem of Algebra (FTA)?

(state similarities and differences)

Similarities: both say the degree equals the

number of zeroes if you count multiplicities

Differences: LFT says each zero

comes from a linear factor.

Linear factorization

)242)(4()( 23 xxxxxf

814872)( 234 xxxxxf

)2)(5.0)(4(2)( 2 xxxxf

)2)(2)(5.0)(4(2)( xxxxxf

This is a “nice” 3rd degree polynomial factor by grouping

What is the difference between:

“k” is a zero of f(x)

(x – k) is a factor of f(x)

Your Turn:

If x = 2, 3, and -4 are zeroes of a function,

write the polynomial in factored form.


)4)(3)(2()( xxxxf

If x = 2i, -2i, and 3 are zeroes of a function,


If x = 3, - 4i, 4i, and -2 are the zeroes of a

function, write the polynomial in factored form.


)3)(2)(2()( xixixxf

)2)(3(44)( xxixixxf

If x = 2i, -2i, and 3 are zeroes of a function,


What is the degree of a polynomial if it has 5 solutions?


)3)(2)(2()( xixixxf

5

Write the function in standard form,identify the zeroes and the x-intercepts

6x

3. To find the standard form polynomial, always multiply

the conjugate pairs first (easier).

)3)(6()( 2 xxxf

1863)( 23 xxxxf

1. The zeroes are: 6x 3ix

)3)(3)(6()( ixixxxf

)3ix

2. The x-intercept is:

(degree 3 polynomial, 3 roots, 1 is real)

Your Turn:

What are the zeroes of the polynomial?

What are the x-intercepts?

)3)(52()52()( xixixxf

3xix 52 ix 52

3x

Remember: a polynomial can be rewritten as a product of linear factors.

814872)( 234 xxxxxf

)2)(2)(5.0)(4(2)( xxxxxf

Complex Conjugates Theorem

If f(x) is a function with real coefficients and if

(a + bi)

is a zero of f(x), then its complex conjugate

(a - bi)

is also a zero of f(x).

Example: x = 1 – 2i, x = 1 + 2i

Irrational Roots Theorem

If f(x) is a function with real coefficients and if

is a zero of f(x), then its irrational conjugate

is also a zero of f(x).

ba

ba

23 ,23 :Example

Describe the end behavior (up left/right, up left down right, etc.)

Positive lead coefficient, even degree

814872)( 234 xxxxxf

Up on left/right

Describe the end behavior (up left/right, up left down right, etc.)

negative lead coefficient, odd degree

81487)( 23 xxxxf

Up on left, down on right

How many real zeroes will the polynomial have?

Does an even degree polynomial necessarily cross

the x-axis?

How many real zeroes will the polynomial have?

Does an odd degree polynomial necessarily cross

the x-axis?

Make a table of the possible zeroes by category

Degree Real zeroes Imaginary Zeroes

2 2 0

1 (mult 2) 0

0 2



3 0 Not possible

1 2

1 mult 2 1 Not possible

3

1 mult 3 0

2 1 Not possible

1, 1 mult 2 0

3 0

If we count multiplicities as separate solutions it is easier.



3 0 3

1 2

2 1

Not possible

Not possible

3 0



4 0

Not possible1 3

2 2

Not possible

4

3 1

4 0

The General Shape of the Graph of a Polynomial

zeroes: x = 2, 3, and -4.

)4)(3)(2()( xxxxf

-4 2 3

All are real numbers. All are x-intercepts.

positive lead coefficient and an odd degree.

The end behavior is ______________________?Up on right, down on left


zeroes: x = 0, -1, 1, and 2.

)2)(1)(1()( xxxxxf

-1 0 1 2


positive lead coefficient and an even degree.

The end behavior is ______________________?Up on right, up on left


zeroes: x = -1, -1, -3, and 4.

)4)(3()1()( 2 xxxxf

-3 -1 4


positive lead coefficient and an even degree.

The end behavior is ______________________?Up on right, up on left

Why doesn’t the “end behavior” line up?


It has the following zeroes: x = -1, -1, -3, and 4.

)4)(3()1()( 2 xxxxf

-3 -1 4

The zero with an EVEN “multiplicity will just

“kiss” the x-axis. Remember ?2xy

Draw the Graph of a Polynomial

It has the following zeroes: x = 2i, -2i, 4, 4, and -2.

)2()4)(2)(2()( 2 xxixixxf

-2 4

Only 4 and -2 are real numbers. These are x-intercepts.positive lead coefficient and an odd degree.

The end behavior is ______________________?Up on right, down on left

The graph “kisses” at x = 4

What is the standard form polynomial?

)3)(52()52()( xixixxf

)3(5252)( xixixxf

ixx 52 xixx 522

)3(5252)( xixixxf

xixx 522

)3(5252)( xixixxf

ix 1042

)3(5252)( xixixxf

225105 iixi ixi 10525

)3(5252)( xixixxf

)3(5252)( xixixxf

)3(5252)( xixixxf

xixx 522 410 2 ix25105 ixi

2942 xx

)3)(294(

)3(5252)(

2

xxx

xixixxf

)3)(294(

)3(5252)(

2

xxx

xixixxf

)294( 2 xxx xxx 294 23

)294(3 2 xx 87123 2 xx

87297 23 xxx

What is the standard form polynomial with the following zeroes?

x = 5, x = 1 – 2i, x = 1 + 2i

1. Write the polynomial in factored form:

)5()21()21()( xixixxf

2. Simplify the “inner” parentheses:

)5(2121)( xixixxf

3. Multiply the conjugate pairs

x = 5, x = 1 – 2i, x = 1 + 2i

)5(2121)( xixixxf

ixx 21 xixx 22

ix 211 xix 21

ixi 212 2422 iixi 422 ixi

4. Combine “like terms”

)5(52)( 2 xxxxf

5. Distributive Property (twice)

x = 5, x = 1 – 2i, x = 1 + 2i

4. Combine “like terms”

)5(52)( 2 xxxxf

52552)( 22 xxxxxxf

25105

52)(

2

23

xx

xxxxf

25157)( 23 xxxxf

math-3 lesson 4-4 roots and graphs of higher degree...

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