Download Document

POLYNOMIAL FUNCTIONS
A POLYNOMIAL is a monomial
or a sum of monomials.
A POLYNOMIAL IN ONE
VARIABLE is a polynomial that
contains only one variable.
Example: 5x2 + 3x - 7
POLYNOMIAL FUNCTIONS
The DEGREE of a polynomial in one variable is
the greatest exponent of its variable.
A LEADING COEFFICIENT is the coefficient
of the term with the highest degree.
What is the degree and leading
coefficient of 3x5 – 3x + 2 ?
POLYNOMIAL FUNCTIONS
A polynomial equation used to represent a
function is called a POLYNOMIAL FUNCTION.
Polynomial functions with a degree of 1 are called
LINEAR POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 2 are called
QUADRATIC POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 3 are called
CUBIC POLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(-2) if f(x) = 3x2 – 2x – 6
f(-2) = 3(-2)2 – 2(-2) – 6
f(-2) = 12 + 4 – 6
f(-2) = 10
POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(2a) if f(x) = 3x2 – 2x – 6
f(2a) = 3(2a)2 – 2(2a) – 6
f(2a) = 12a2 – 4a – 6
POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(m + 2) if f(x) = 3x2 – 2x – 6
f(m + 2) = 3(m + 2)2 – 2(m + 2) – 6
f(m + 2) = 3(m2 + 4m + 4) – 2(m + 2) – 6
f(m + 2) = 3m2 + 12m + 12 – 2m – 4 – 6
f(m + 2) = 3m2 + 10m + 2
POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find 2g(-2a) if g(x) = 3x2 – 2x – 6
2g(-2a) = 2[3(-2a)2 – 2(-2a) – 6]
2g(-2a) = 2[12a2 + 4a – 6]
2g(-2a) = 24a2 + 8a – 12
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = 3
Constant
Function
Degree = 0
Max. Zeros: 0
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x + 2
Linear
Function
Degree = 1
Max. Zeros: 1
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
Quadratic
Function
Degree = 2
Max. Zeros: 2
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
Cubic
Function
Degree = 3
Max. Zeros: 3
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
Quartic
Function
Degree = 4
Max. Zeros: 4
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1
Quintic
Function
Degree = 5
Max. Zeros: 5
POLYNOMIAL FUNCTIONS
END BEHAVIOR
f(x) = x2
Degree: Even
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  +∞
POLYNOMIAL FUNCTIONS
END BEHAVIOR
f(x) = -x2
Degree: Even
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  -∞
POLYNOMIAL FUNCTIONS
END BEHAVIOR
f(x) = x3
Degree: Odd
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  +∞
POLYNOMIAL FUNCTIONS
END BEHAVIOR
f(x) = -x3
Degree: Odd
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  -∞
Complex Numbers
i  1
2
Note that squaring both sides yields: i  1
therefore i 3  i 2 * i1  1* i  i
and i 4  i 2 * i 2  (1) * (1)  1
so
and
i  i * i  1* i  i
5
4
i  i * i  1* i  1
6
4
And so on…
2
2
Real numbers and imaginary numbers are
subsets of the set of complex numbers.
Real Numbers
Imaginary
Numbers
Complex Numbers
Definition of a Complex Number
If a and b are real numbers, the number a + bi is a
complex number, and it is said to be written in
standard form.
If b = 0, the number a + bi = a is a real number.
If a = 0, the number a + bi is called an imaginary
number.
Write the complex number in standard form
1   8  1  i 8  1  i 4  2  1  2i 2
Addition and Subtraction of Complex
Numbers
If a + bi and c +di are two complex numbers written
in standard form, their sum and difference are
defined as follows.
Sum: ( a  bi )  ( c  di )  ( a  c )  ( b  d
)i
Difference:( a  bi )  ( c  di )  ( a  c )  ( b  d
)i
Perform the subtraction and write the answer
in standard form.
( 3 + 2i ) – ( 6 + 13i )
3 + 2i – 6 – 13i
–3 – 11i
8   18  4  3i 2 
8  i 9  2  4  3i 2 
8  3i 2  4  3i 2
4
Multiplying Complex Numbers
Multiplying complex numbers is similar to
multiplying polynomials and combining like terms.
Perform the operation and write the result in standard
form. ( 6 – 2i )( 2 – 3i )
F
O
I
L
12 – 18i – 4i + 6i2
12 – 22i + 6 ( -1 )
6 – 22i
The Fundamental Theorem of Algebra
We have seen that if a polynomial equation is of degree n, then counting
multiple roots separately, the equation has n roots. This result is called the
Fundamental Theorem of Algebra.
The Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n, where n  1, then the equation f(x)  0 has
at least one complex root.
The Linear Factorization Theorem
Just as an nth-degree polynomial equation has n roots, an nth-degree
polynomial has n linear factors. This is formally stated as the Linear
Factorization Theorem.
The Linear Factorization Theorem
If f(x)  anxn  an1xn1  …  a1x  a0 b, where n  1 and an  0 , then
f (x)  an (x  c1) (x  c2) … (x  cn)
where c1, c2,…, cn are complex numbers (possibly real and not necessarily
distinct). In words: An nth-degree polynomial can be expressed as the product
of n linear factors.
Find all the zeros of f ( x)  x5  2 x 4  8x 2  13x  6
Solutions:
The possible rational zeros are 1, 2, 3, 6
Synthetic division or the graph can help:
Notice the real zeros appear as xintercepts. x = 1 is repeated zero
since it only “touches” the x-axis,
but “crosses” at the zero x = -2.
50
40
30
20
10
-4
-3
-2
-2
1
-1
1
-10
-20
-30
-40
-50
2
3
4
Thus 1, 1, and –2 are
real zeros. Find the
remaining 2 complex
zeros.
Write a polynomial function f of least degree that has real
coefficients, a leading coefficient 1, and 2 and 1 + i as zeros).
Solution:
f(x) = (x – 2)[x – (1 + i)][x – (1 – i)]
 x3  4 x 2  6 x  4
Factoring Cubic Polynomials
Find the Greatest Common
Factor
14x3 – 21x2
2•7•x•x•x
GCF = 7x2
–
3•7•x•x
Identify each term in the polynomial.
Identify the common factors in each
term
The GCF is?
14x3 – 21x2 = 7x2(2x – 3)
Use the distributive property to
factor out the GCF from each
term
Factor Completely
4x3 + 20x2 + 24x
2•2•x•x•x
+
Identify each term in the polynomial.
2•2•5•x•x
+
2•2•2•3•x
GCF = 4x
Identify the common
factors in each term
The GCF is?
4x3 + 20x2 + 24x = 4x(x2 + 5x +6)
4x (x + 2)(x + 3)
Use the distributive
property to factor out
the GCF from each
term
Factor by Grouping
x3 - 2x2 - 9x + 18 Group terms in the polynomial.
Identify a common
= (x3 - 2x2) + (- 9x + 18)
factor in each group
x•x•x-2•x•x + -3•3•x+2•3•3
and factor
= x2(x – 2) + -9(x – 2)
= (x – 2)(x2 – 9)
Now identify the common
factor in each term
Use the distributive property
= (x – 2)(x – 3)(x + 3)
Factor the difference of two squares
Sum of Two Cubes Pattern
a3 + b3 = (a + b)(a2 - ab + b2)
Example
x3 + 27 = x3 + 3•3•3 = x3 + 33
x3 + 33 = (x + 3)(x2 - 3x + 32)
= (x + 3)(x2 - 3x + 9)
So x3 + 27 = (x + 3)(x2 - 3x + 9)
Now, use the
pattern to factor
Difference of Two Cubes Pattern
a3 - b3 = (a - b)(a2 + ab + b2)
Example
n3 - 64 = n3 - 4•4•4 = n3 - 43
n3 - 43 = (n - 4)(n2 + 4n + 42)
= (n - 4)(n2 + 4n + 16)
So n3 - 64 = (n - 4)(n2 + 4n + 16)
Now, use the
pattern to factor