Lesson 4-1 Polynomial Functions.

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Presentation transcript:

Lesson 4-1 Polynomial Functions

Vocabulary Polynomial 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) = 0 Imaginary Number – a complex number of the form where and is the imaginary number. Complex numbers – the imaginary numbers combined with the real numbers Pure 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 3 Determine whether -2 is a zero of f(x) f(-2) = 59 Not a zero

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

State the degree and the leading coefficient of the polynomial Degree of 5 and leading coefficient of 8 Determine whether 1 is a zero of f(x) f(1) = 15 No

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

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

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

State the number of complex roots of the equation has State 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 roots

Graph with the graphing Calculator

Graph on 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 calculator

Total of 3 Complex Roots, 2 imaginary, 1 real

Graph on graphing calculator 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 calculator

Total of 2 Complex Roots, 2 real (double root)

Graph on 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 calculator

3 complex roots 3 real roots x=0, x=-4, x=2

Example 5