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Pre-Calculus Chapter Four Section 1 Notes
Polynomials
The degree of a polynomial in one variable is the greatest exponent of its variable.
The coefficient of the variable with the greatest exponent is called the leading coefficient.
Example 1:
Consider the following polynomial:
f ( x)  3 x 3  2 x 2  3 x  4
Degree: ______
Leading coefficient: _______
Zeros of Polynomials:
If P is a polynomial and if c is a number such that P(c)=0, then we say that c
is a zero of P. The following are equivalent ways of saying the same thing.
1.
2.
3.
4.
c is a zero of P.
x = c is a root of the equation P(x)=0.
x – c is a factor of P(x).
x = c is an x-intercept of the graph of P.
Example 2: Determine if each number is a root of
When x = 0 , x = 2, and x = -3/2
f ( x)  2 x 4  x3  3x 2
Imaginary Numbers!
x2  4  0
Consider
, until now, we couldn’t find a solution for x which makes this
true. However, in the complex number system, we can find a value for x that satisfies
the equation.
Example 3: solve the equation.
x2  4  0
So x   4i
A root or zero of P(x) may also be an imaginary number such as bi.
Definition:
i  1
i 2  1
i3   1
i4  1
i5  1
A complex number is an expression of the form
a + bi
where a and b are real numbers and i 2 = -1 . The real part of this complex number a and
the imaginary part is b. Two complex numbers are equal if and only if their real parts
are equal and their imaginary parts are equal.
If b = 0 then the complex number is a real number, if a = 0 and b does not equal zero,
then the complex number is called a pure imaginary number.
Fundamental Theorem of Algebra
Every polynomial equation with degree greater than zero has at least one root in the set of
complex numbers.
Even or odd
The graph of a polynomial function with odd degree must cross the x-axis at least once.
The graph of a function with even degree may or may not cross the x-axis. If it does, it
will cross an evem number of times.
Example 4:
A. Write a polynomial equation of least degree with roots 2 , 3i , -3i.
B. Does the equation have an odd or even degree? How many times does the graph of the related
function cross the x-axis?
Example 5:
A. State the number of complex roots of the equation
find the roots and graph the related function.
32 x3  32 x 2  4 x  4  0 . The