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Polynomial and Rational Functions. Aim #2.1 What are complex numbers?. The imaginary unit I is defined as. Complex numbers and Imaginary Numbers. The set of all numbers in the form of a + bi , with real numbers a and b and i , the imaginary unit, is called the set of
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Aim #2.1 What are complex numbers? • The imaginary unit I is defined as
Complex numbers and Imaginary Numbers • The set of all numbers in the form of a + bi, with real numbers a and b and i, the imaginary unit, is called the set of complex numbers. • An imaginary number in the form of bi is called a pure imaginary number. • Example: -4 + 6i, 0+ 2i= 2i
A complex number is said to be simplified if its in the form of a + bi. • If b contain a radical express i before the radical.
Equality of Complex numbers: a + bi= c + di if and only if a = c and b = d
(5 – 11i) + (7 + 4i) Steps: Add or subtract the real parts Add or subtract the imaginary parts Express final answer as a complex number Operations with Complex Numbers
(-5 + i) – (-11 – 6 i) Steps: Add or subtract the real parts Add or subtract the imaginary parts Express final answer as a complex number Operations with Complex Numbers
4i(3- 5i) Distribute 4i throughout the parenthesis Multiply Replace i2 with -1. Simplify Multiplying Complex Numbers
(7 - 3i)(-2- 5i) Use the Foil Method or Vertical Method to multiply Replace i2 with -1 Simplify Multiplying Complex Numbers
What are Conjugates? • The complex conjugate of the numbera + biisa – bi and vice versa. • When you multiply a complex number by its conjugate you get a real number.
Divide and express the result in standard form. Multiply the numerator and denominator by the denominators conjugate. Use FOIL (or the Vertical Method) Replace i2 with -1 Simplify Express final answer in standard form. Using Complex Conjugates
Roots of Negative Numbers • Perform the indicated operation.
Summary: Answer in complete sentences. • What is i? • Explain how to add or subtract complex numbers. • What is the conjugate of a complex number? • Explain how to divide complex numbers and provide an example.
Aim #2.2: What are some properties of quadratic functions? • The Standard Form of Quadratic Function:
General Form of a Quadratic: • f (x)= ax2 + bx + c • To find the vertex using this form you need the axis of symmetry:
Graphing a Quadratic Function • Identify which way the parabola will open • Identify the vertex • Find the y –intercept by evaluating f(0). • Find the x-intercepts. • Then graph.
Converting from General to Standard Form • Convert the function: • Y = x2 + 4x – 1 • Steps:
Practice: • Convert to standard form. Y = 3x2 + 6 x + 7
Summary:Answer in complete sentences. 3- List three things you learned about quadratic functions. 2- List 2 ways you can apply this to real world. 1- Write one question that you may still have on this topic.
Aim #2.3: How do we identify polynomial functions and their graphs? • Examples and Non examples:
Smooth, Continuous Graphs • Polynomial functions of degree 2 or higher have graphs that are smooth and continuous.
Use the leading coefficient test to determine the end behavior. f (x)= x3 + 3x2 – x - 3 End behavior is how we describe a graph to the far left or far right. Using the Test: The leading coefficient is 1. The exponent is odd Odd-degree have graphs with opposite behavior at each end. Leading Coefficient Test
Leading Coefficient Test • Odd – degree; positive leading coefficient • Graphs falls left and increases right • Odd – degree; negative leading coefficient • Graph rises left and falls right
Use the leading coefficient test to determine the end behavior. f (x)= x4 – 4x2 Even degree; positive leading coefficient Rises left and rises right Even degree; negative leading coefficient Falls left and falls right Guided Practice:
Using the Leading Coefficient Test • Use the leading coefficient test to determine the end behavior of the graph of: • f (x) = -4x3 ( x -1)2 (x + 5)
Zeros of Polynomial Functions • Find all the zeros: f (x)= x3 + 3x2 –x -3
Practice: • Find all the zeros: f (x)= x3 + 2x2 –4x -8
Find all the zeros: f (x)= - x4 + 4x3 – 4x2 Steps: Set f (x) = 0 Multiply both sides by -1. Factor the GCF. Factor completely. Solve for x. Finding Zeros of a Polynomial Function
If ris a zero with even multiplicity, then the graph touches the x-axis and turns around at r. Multiplicity and X-intercepts
Multiplicity and X-intercepts • If ris a zero with odd multiplicity, then the graph crosses the x-axis at r. • Note: Regardless of multiplicity graphs tend to flatten out near the zeros with multiplicity greater than one.
Finding Zeros and their Multiplicities • Find the zeros of f (x)= ½ (x + 1) (2x – 3)2 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. • Steps: 1. Set f (x)= 0 and set each variable factor to 0.
Aim #2.5: How do we find the zeros of a polynomial function? • Rational Zero Theorem provides us with a tool we can use to make a list of all possible rational zeros of a polynomial function.
