Math 1330 2.2
2.2 Polynomials
Definition of a Polynomial Function
Let n be a nonnegative integer and let a n , a n 1 ,..., a 2 , a1 , a 0 , be real numbers, with a n  0 .
The function defined by f ( x )  a n x n  ...  a 2 x 2  a1 x  a 0 is called a polynomial function of x of degree n.
The number a n , the coefficient of the variable to the highest power, is called the leading coefficient.
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For example, p ( x )  2 x 3  3 x 10  6 x 0.234 is NOT a polynomial, but a ( x )  2 x 3  x 10  0.234 x is a
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polynomial.
The domain of any polynomial function is all real numbers.
End Behavior of Polynomial Functions
The behavior of a graph of a function to the far left or far right is called its end behavior.
Even-degree
LEADING COEFFIECIENT: +
Both ends up
LEADING COEFFICIENT: -
Both ends down
Odd-degree
LEADING COEFFIECIENT: +
LEADING COEFFICIENT: -
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Math 1330 2.2
Rising Left to right
Fall in Left to right
Power functions:
A power function is a polynomial that takes the form f ( x)  ax n , where n is a positive integer. Modifications of
power functions can be graphed using transformations.
Even-degree power functions:
Odd-degree power functions:
Note: Multiplying any function by a will multiply all the y-values by a. The general shape will stay the same.
Here is an example of a polynomial function:
Zeros of polynomials:
If f is a polynomial and c is a real number for which f (c )  0 , then c is called a zero of f, or a root of f.
If c is a zero of f, then
 c is an x-intercept of the graph of f.
 ( x  c ) is a factor of f.
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Math 1330 2.2
So if we have a polynomial in factored form, we know all of its x-intercepts.
 every factor gives us an x-intercept.
 every x-intercept gives us a factor.
Example 1: Consider the function: f ( x )   2( x  3 )( x  4 )( 2  x )
Zeros (x-intercepts):
y-intercept:
leading term and degree:
Steps to graphing other polynomials:
(Note: this approach is completely different from the method in the textbook!)
1. Factor and find x-intercepts.
2. Mark x-intercepts on x-axis.
3. Determine the leading term.

Degree: is it odd or even?

Sign: is the coefficient positive or negative?
4. Determine the end behavior. What does it “look like”?
Odd Degree
Sign (+)
Odd Degree
Sign (-)
Even Degree
Sign (+)
Even Degree
Sign (-)
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Math 1330 2.2
5. For each x-intercept, determine the behavior.

Even multiplicity: touches x-axis, but doesn’t cross (looks like a parabola there).

Odd multiplicity of 1: crosses the x-axis (looks like a line there).

Odd multiplicity  3 : crosses the x-axis and looks like a cubic there.
6. Draw the graph, being careful to make a nice smooth curve with no sharp corners.
Example 2: Find the x and y intercepts of the graph of the function. State the degree of the function.
P( x)   x  3 x  1x  2 . Then sketch the graph of the function, labeling all intercepts. Show the correct
behavior at each x intercept and show the proper end behavior.
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Math 1330 2.2
Example 3: Find the x and y intercepts of the graph of the function. State the degree of the function.
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P ( x )   x  1  x  2  x  1 . Then sketch the graph of the function, labeling all intercepts. Show the correct
behavior at each x intercept and show the proper end behavior.
Example 4: Write the equation of the cubic polynomial P(x) that satisfies the following conditions: zeros at
x  3, x  1, and x  4 and passes through the point (-3, 7).
Example 5: Write the equation of the quartic function with y intercept 4 which is tangent to the x axis at the
points (-1, 0) and (1, 0).
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Math 1330 2.2
Example 6: Find the equation for the following graph.
Example 7: Find the equation for the following graph.
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Math 1330 2.2
Example 8: Find the equation for the following graph.
Example 9: Graph the following function:
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