Slide 1

Lesson 4-1 Polynomial Functions VocabularyPolynomial in one variable an expression of the form . The coefficients represent complex numbers (real or imaginary) is not zero, and n represents a nonnegative integer.Degree the greatest exponent of a polynomials variable.Leading coefficient the coefficient of the variable with the greatest exponent.Polynomial function a function where P(x) is a polynomial in one variable.

Polynomial equation - a polynomial that is set equal to zero.Root a solution of the equation P(x) = 0Imaginary Number a complex number of the form where and is the imaginary number.Complex numbers the imaginary numbers combined with the real numbersPure Imaginary Number the complex number when a = 0 and Fundamental Theorem of Algebra Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers

Polynomial Functions

State the degree and the leading coefficient of the polynomial

Degree of 4 and leading coefficient of 3Determine whether -2 is a zero of f(x) f(-2) = 59 Not a zeroState the degree and the leading coefficient of the polynomial

Degree of 3 and leading coefficient of 1Determine whether 6 is a zero of f(x) f(6) = 0 Yes it is a zeroState the degree and the leading coefficient of the polynomial

Degree of 5 and leading coefficient of 8Determine whether 1 is a zero of f(x) f(1) = 15 NoExample 2

Fundamental Theorem of Algebra

Corollary to the Fundamental Theorem of Algebra

Example 3

Write a polynomial equation of least degree with roots 2, 3i, and -3i

Write a polynomial equation of least degree with roots -5 and 7

Write a polynomial equation of least degree with roots 6, 2i, -2i, i,-i

Example 3

Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis.

Odd, onceDoes the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis.

Even, twiceDoes the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis.

Odd, onceState the number of complex roots of the equation has. Then find the roots and graph the related function.

The polynomial has a degree of 4 so there are 4 complex roots.

2 real roots and 2 imaginary roots for a total of 4 complex rootsGraph with the graphing Calculator

State the number of complex roots of the equation has. Then find the roots and graph the related function.

3 Complex Roots

Graph on graphing calculatorTotal of 3 Complex Roots,2 imaginary, 1 real

x-1=0x=1

State the number of complex roots of the equation has. Then find the roots and graph the related function.

2 Complex Roots

Graph on graphing calculatorTotal of 2 Complex Roots,2 real (double root)x-7=0x=7

State the number of complex roots of the equation has. Then find the roots and graph the related function.

3 Complex Roots

Graph on graphing calculator3 complex roots3 real roots

x=0, x=-4, x=2Example 5