Section 2.1 Complex Numbers. The Imaginary Unit i

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?