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.
))...()(()( 21 nzxzxzxaxf
)4)(3)(2()( xxxxf
If x = 2i, -2i, and 3 are zeroes of a function,
write the polynomial in factored form.
If x = 3, - 4i, 4i, and -2 are the zeroes of a
function, write the polynomial in factored form.
))...()(()( 21 nzxzxzxaxf
)3)(2)(2()( xixixxf
)2)(3(44)( xxixixxf
If x = 2i, -2i, and 3 are zeroes of a function,
write the polynomial in factored form.
What is the degree of a polynomial if it has 5 solutions?
))...()(()( 21 nzxzxzxaxf
)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
Make a table of the possible zeroes by category
Degree Real zeroes Imaginary Zeroes
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.
Make a table of the possible zeroes by category
Degree Real zeroes Imaginary Zeroes
3 0 3
1 2
2 1
Not possible
Not possible
3 0
Make a table of the possible zeroes by category
Degree Real zeroes Imaginary Zeroes
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
The General Shape of the Graph of a Polynomial
zeroes: x = 0, -1, 1, and 2.
)2)(1)(1()( xxxxxf
-1 0 1 2
All are real numbers. All are x-intercepts.
positive lead coefficient and an even degree.
The end behavior is ______________________?Up on right, up on left
The General Shape of the Graph of a Polynomial
zeroes: x = -1, -1, -3, and 4.
)4)(3()1()( 2 xxxxf
-3 -1 4
All are real numbers. All are x-intercepts.
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?
The General Shape of the Graph of a Polynomial
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