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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
13)( 234 xxxxf
Degree of 4 and leading coefficient of 3Determine whether -2 is a zero of f(x) f(-2) = 59 Not a zero
State the degree and the leading coefficient of the polynomial
1835)( 23 xxxxf
Degree of 3 and leading coefficient of 1Determine 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
285)( 52 xxxf
Degree of 5 and leading coefficient of 8Determine 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
01892
0)9)(2(
0)9)(2(
0)3)(3)(2(
23
2
22
xxx
xx
ixx
ixixx
Write a polynomial equation of least degree with roots -5 and 7
0352
0)7)(5(2
xx
xx
Write a polynomial equation of least degree with roots 6, 2i, -2i, i,-i
02443056
0)45)(6(
0)1)(4)(6(
0))(4)(6(
0))()(2)(2)(6(
2345
24
22
2222
xxxxx
xxx
xxx
ixixx
ixixixixx
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.
01892 23 xxx
Odd, once
Does the equation have an odd or even degree? How many times does the graph of
the related function cross the x-axis.
03522 xxEven, twice
Does the equation have an odd or even degree? How many times does the graph of
the related function cross the x-axis.
02443056 2345 xxxxx
Odd, once
State the number of complex roots of the equation has. Then find the roots and
graph the related function.
04359 24 xx
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
04359 24 xx
State the number of complex roots of the equation has. Then find the roots and
graph the related function.
0443232 23 xxx
3 Complex Roots
Graph on graphing calculator
Total of 3 Complex Roots,2 imaginary, 1 real
4
2
32
4
432
)1)(432(
0)1(4)1(32
2
2
2
2
ix
x
x
xx
xxx
x-1=0x=1
0443232 23 xxx
State the number of complex roots of the equation has. Then find the roots and
graph the related function.
049142 xx
2 Complex Roots
Graph on graphing calculator
Total of 2 Complex Roots,2 real (double root)
x-7=0x=77
0)7(
)7)(7(
049142
x
x
xx
xx049142 xx
State the number of complex roots of the equation has. Then find the roots and
graph the related function.
082 23 xxx
3 Complex Roots
Graph on graphing calculator
3 complex roots3 real roots
0)2)(4(
0)82(
0822
23
xxx
xxx
xxx
x=0, x=-4, x=2
Example 5