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Lesson 4-1 Polynomial Functions
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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.
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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
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Polynomial Functions
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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
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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
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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
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Example 2
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Fundamental Theorem of Algebra
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Corollary to the Fundamental Theorem of Algebra
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Example 3
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Write a polynomial equation of least degree with roots 2, 3i, and -3i
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Write a polynomial equation of least degree with roots -5 and 7
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Write a polynomial equation of least degree with roots 6, 2i, -2i, i,-i
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Example 3
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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
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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
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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
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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.
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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
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Graph with the graphing Calculator
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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
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Total of 3 Complex Roots, 2 imaginary, 1 real
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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
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Total of 2 Complex Roots, 2 real (double root)
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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
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3 complex roots 3 real roots
x=0, x=-4, x=2
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Example 5
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