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Section 2.1Complex Numbers
The Imaginary Unit i
2
The Imaginary Unit
The imaginary unit is defined as
= -1, where 1.
i
i
i i
Complex Numbers and Imaginary Numbers
The set of all numbers in the form
a+b
with real numbers a and b, and i, the imaginary unit,
is called the set of complex numbers. The real number
a is called the r
i
eal part and the real number b is called
the imaginary part of the complex number a+b . If b 0,
then the complex number is called an imaginary number.
An imaginary number in the form b is called a p
i
i
ure
imaginary number.
Equity of Complex Numbers
a+b =c+d if and only if a=c and b=d.i i
Example
Express as a multiple of i:
2
16
7i
Operations with
Complex Numbers
Adding and Subtracting Complex Numbers
1. a+b d = a+c b+d
This says that you add complex numbers by adding their real
parts, adding their imaginary parts, and expressing the sum as
a complex number.
2
i c i i
. a+b c+d a-c -d
This says that you subtract complex numbers by subtracting
their real parts, subtracting their imaginary parts, and
expressing the difference as a complex number.
i i b i
Example
Perform the indicated operation:
7 4 9 5
8 3 17 7
i i
i i
Multiplication of complex numbers is
performed the same way as multiplication
of polynomials, using the distributive
property and the FOIL method.
Example
Perform the indicated operation:
3 5 6 2i i
Complex Conjugates
and Division
2 2
Conjugate of a Complex Number
The complex conjugate of the number a+bi is a-bi,
and the complex conjugate of - is . The
multiplication of complex conjugates gives a real
number.
a bi a bi
a bi a bi a b
a bi
2 2a bi a b
Using complex conjugates to divide complex numbers
Example
Divide and express the result in standard form:
7 6
5 9
i
i
Roots of Negative Numbers
Because the product rule for radicals only
applies to real numbers, multiplying radicands
is incorrect. When performing operations
with square roots of negative numbers, begin
by expressing all square roots in terms of i.
Then perform the indicated operation.
Principal Square Root of a Negative Number
For any positive real number b, the principal square
root of the negative number -b is defined by
-b i b
Example
Perform the indicated operations and write the result in standard form:
54 7 24
Example
Perform the indicated operations and write the result in standard form:
2
4 7
Section 2.2Quadratic Functions
Graphs of Quadratic Functions
Graphs of Quadratic Functions Parabolas
x
y
x
y
MinimumVertex
Axis of symmetry Maximum
2( )f x ax bx c
2
Quadratic functions are any function of the form
f(x)=ax +bx+c where a 0, and a,b and c are
real numbers. The graph of any quadratic
function is called a parabola. Parabolas are
shaped like cups. Para
bolas are symmetic with
respect to a line called the axis of symmetry.
If a parabola is folded along its axis of symmetry,
the two halves match exactly.
Graphing Quadratic Functions in Standard Form
Seeing the Transformations
Example
Graph the quadratic function f(x) = - (x+2)2 + 4.
x
y
Graphing Quadratic Functions in the Form f(x)=ax2=bx+c
2
We can identify the vertex of a parabola whose equation is in
the form f(x)=ax +bx+c. First we complete the square.
Using the form f(x)=ax2+bx+c
2
Finding y intercept
y=0 2 0 1
1 (0,1) y intercepty
x
y
2( ) 2 1 a=1, b=2, c=1f x x x
2
-b -2, x= 1
2a 2 2 1
( 1) ( 1) 2( 1) 1 0 V(-1,0)
bVertex f
a
f
Axis of symmetry x=-1
2
Finding x intercept
0=x 2 1
0 ( 1)( 1)
1 0
1 (-1,0) x intercept
x
x x
x
x
a>0 so parabola has a minimum, opens up
Example
Find the vertex of the function f(x)=-x2-3x+7
Example
Graph the function f(x)= - x2 - 3x + 7. Use the graph to identify the domain and range.
x
y
Minimum and Maximum Values of Quadratic Functions
Example
For the function f(x)= - 3x2 + 2x - 5
Without graphing determine whether it has a minimum or maximum and find it.
Identify the function’s domain and range.
Graphing Calculator – Finding the Minimum or Maximum
Input the equation into Y=
Go to 2nd Trace to get Calculate. Choose #4 for Maximum or #3 for Minimum.
Move your cursor to the left (left bound) of the relative minimum or maximum point that you want to know the vertex for. Press Enter.
Then move your cursor to the other side of the vertex – the right side of the vertex when it asks for the right bound. Press Enter.
When it asks to guess, you can or simply press Enter.
The next screen will show you the coordinates of the maximum or minimum.
Section 2.3Polynomial Functions and
Their Graphs
Smooth, Continuous Graphs
Polynomial functions of degree 2 or higher have graphs that are smooth and continuous. By smooth, we mean that the graphs contain only rounded curves with no sharp corners. By continuous, we mean that the graphs have no breaks and can be drawn without lifting your pencil from the rectangular coordinate system.
Notice the breaks and lack of smooth curves.
End Behavior of Polynomial Functions
Odd-degree polynomial
functions have graphs with
opposite behavior at each end.
Even-degree polynomial
functions have graphs with the
same behavior at each end.
Example
Use the Leading Coefficient Test to determine the end behavior of the graph of f(x)= - 3x3- 4x + 7
Zeros of Polynomial Functions
If f is a polynomial function, then the values of x for which f(x) is equal to 0 are called the zeros of f. These values of x are the roots, or solutions, of the polynomial equation f(x)=0. Each real root of the polynomial equation appears as an x-intercept of the graph of the polynomial function.
Find all zeros of f(x)= x3+4x2- 3x - 12
3 2
2
2
2
By definition, the zeros are the values of x
for which f(x) is equal to 0. Thus we set
f(x) equal to 0 and solve for x as follows:
(x 4 ) (3 12) 0
x (x 4) 3(x 4) 0
x+4 x - 3 0
x+4=0 x - 3=0
x=-4
x x
2 x 3
x = 3
Example
Find all zeros of x3+2x2- 4x-8=0
Multiplicity of x-Intercepts
2 2For f(x)=-x ( 2) , notice that each
factor occurs twice. In factoring this equation
for the polynomial function f, if the same
factor x- occurs times, but not +1 times,
we call a zero with multip
x
r k k
r
licity . For the
polynomial above both 0 and 2 are zeros with
multiplicity 2.
k
3 2
3 2
2
2
Find the zeros of 2 4 8 0
2 4 8 0
x 2 4( 2) 0
2 4 0
x x x
x x x
x x
x x
2 2 2 0
2 has a multiplicity of 2, and 2 has a multiplicity of 1.
Notice how the graph touches at -2 (even multiplicity),
but crosses at 2 (odd multiplicity).
x x x
Graphing Calculator- Finding the Zerosx3+2x2- 4x-8=0
One of the zeros
The other zero
Other zero
One zero of the function
The x-intercepts are the zeros of the function. To find the zeros, press 2nd Trace then #2. The zero -2 has multiplicity of 2.
Example
Find the zeros of f(x)=(x- 3)2(x-1)3 and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.
Continued on the next slide.
Example
Now graph this function on your calculator. f(x)=(x- 3)2(x-1)3
x
y
The Intermediate Value Theorem
Show that the function y=x3- x+5 has a zero between - 2 and -1.
3
3
f(-2)=(-2) ( 2) 5 1
f(-1)=(-1) ( 1) 5 5
Since the signs of f(-1) and f(-2) are opposites then
by the Intermediate Value Theorem there is at least one
zero between f(-2) and f(-1). You can also see th
ese values
on the table below. Press 2nd Graph to get the table below.
Example
Show that the polynomial function f(x)=x3- 2x+9 has a real zero between - 3 and - 2.
Section 2.4Dividing Polynomials;
Remainder and Factor Theorems
Long Division of Polynomials and
The Division Algorithm
Dividing Polynomials Using
Synthetic Division
The Factor Theorem
Solve the equation 2x3-3x2-11x+6=0 given that 3 is a zero of f(x)=2x3-3x2-11x+6. The factor theorem tells us that x-3 is a factor of f(x). So we will use both synthetic division and long division to show this and to find another factor.
Another factor
Example
Solve the equation 5x2 + 9x – 2=0 given that -2 is a zero of f(x)= 5x2 + 9x - 2
Section 2.5Zeros of Polynomial Functions
The Rational Zero Theorem
Example
List all possible rational zeros of f(x)=x3-3x2-4x+12
Find one of the zeros of the function using synthetic division, then factor the remaining polynomial. What are all of the zeros of the function? How can the graph below help you find the zeros?
4 3
Notice that the roots for our most recent problem
(x -x 7 9 18 0; degree 4) were 3i,2,-1x x
The Fundamental Theorem of Algebra
Remember that having roots of 3, -2, etc. are
complex roots because 3 can be written 3+0i
and -2 can be written as -2+0i.
The Linear Factorization Theorem
Example
Find a fourth-degree polynomial function f(x) with real coefficients that has -1,2 and i as zeros and such that f(1)=- 4
Descartes’s Rule of Signs
Example
For f(x)=x3- 3x2- x+3 how many positive and negative zeros are there? What are the zeros of the function?