Unit 5 – Higher Order Polynomials
Polynomial Vocabulary:
 Polynomial –
Name: ____________________________________
Characteristics of Polynomials and their Graphs

_____________________ of the polynomial - highest power, determines the total number of __________________

Leading ___________________ - coefficient of and variable with highest power

Leading _______________________________ - coefficient of the leading term
Odd
Degree
Even
End Behavior: what happens to the value of the _______________________________________ as x values approach −∞ and ∞.
Sign of Leading Coefficient
Positive
Negative
𝑎𝑠 𝑥 → ∞, 𝑦 → ______
𝑎𝑠 𝑥 → −∞, 𝑦 → ______
𝑎𝑠 𝑥 → ∞, 𝑦 → ______
𝑎𝑠 𝑥 → −∞, 𝑦 → ______
𝑎𝑠 𝑥 → ∞, 𝑦 → ______
𝑎𝑠 𝑥 → −∞, 𝑦 → ______
Forms of Polynomials Equations/Functions:
 Separate Terms:
𝑎𝑠 𝑥 → ∞, 𝑦 → ______
𝑎𝑠 𝑥 → −∞, 𝑦 → ______

Factored: 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 4)(𝑥 2 + 9)
Characteristics of Graphs
1. Turns:
 points where the graph changes from
(walking uphill) to __________________________________
(walking downhill) and vice versa
 happen at maxs and mins
 always less than the ______________________
2. X-intercepts – where the graph __________________________ OR __________________________ the x-axis
Double x-intercepts
 same solution ________________________ when solving
 double factor
 graph only ______________________________, does not ______________________ x-axis
Imaginary x-intercepts
 always in ____________________________
 X-intercept rules:
 Even degree
 Odd degree:
o 0 x-intercepts OR
o 1 x-intercept OR
o Any ______________ number ____________ the degree
o Any ______________ number ____________ the degree
3. Y-intercept – evaluate the polynomial at x = 0
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Variations of Common Polynomials
Quadratic (𝒚 = 𝒙𝟐 ) (Degree 2)
Two real
Two imaginary
Linear (𝒚 = 𝒎𝒙 + 𝒃) (Degree 1)
One real root
y = ___________________________
y = __________________________ y = ________________________
One real (double root)
y = ______________________
Cubic (𝒚 = 𝒙𝟑 ) (Degree 3)
Three real roots:
One real / two imaginary roots:
One real root (triple root):
y = ___________________________
y = ___________________________
y = ___________________________
Quartic (𝒚 = 𝒙𝟒) (Degree 4)
Four real roots:
Two real / two imaginary:
Four imaginary roots:
One real (quadruple root):
y = ___________________________ y = ___________________________ y = ___________________________ y = ___________________________
Quintic (𝒚 = 𝒙𝟓 ) (degree 5)
Five real roots:
3 real roots/2 imag:
1 real (4 imag):
y = __________________________
y = ___________________________
y = ___________________________
Without using a graphing calculator, match the equations with the correct graph.
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𝑓(𝑥) = 3𝑥 3 − 8𝑥 2 + 4𝑥
𝑔(𝑥) = −𝑥 3 + 2𝑥 2 − 3
𝑘(𝑥) = −3𝑥 4 + 5𝑥 3 − 4𝑥 + 1
ℎ(𝑥) = 𝑥 4 − 2𝑥 2 + 1
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Roots: ___________________
# of Turns: _____________
Roots: ___________________
# of Turns: _____________
Roots: ___________________
# of Turns: _____________
Roots: ___________________
# of Turns: _____________
y-int: ___________________
y-int: ___________________
y-int: ___________________
y-int: ___________________
End behavior: _________
End behavior: ________
End behavior: ________
End behavior: ________
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Finding Roots/Solving Polynomials
More Polynomial Vocabulary:
𝑥 = 1 is a
𝒇(𝒙) = 𝟑𝒙𝟐 + 𝟒𝒙 − 𝟕
of 𝑓(𝑥).
𝑥 = 1 is a
of 𝑓(𝑥) because 𝑓(1) = 0.
(𝑥 − 1) is a
(1,0) is a
of 𝑓(𝑥).
of the graph of 𝑓(𝑥).
To solve/find the roots of a polynomial:
1. Take out 𝑓(𝑥) and put in 0 OR move all terms to one side and get 0 on the other
2. Use one of the methods based on the degree of the polynomial.
3. Write answers as ordered pairs: ___________________________

Methods for Degree 2 ONLY
1.
Quadratic Formula

2. Taking the √
3. Difference of Two Squares
Methods for Degree 3 ONLY
1. Sum of Cubes


of Both Sides
2. Difference of Cubes
Methods for Degree 2 and 4 Trinomials
1. Degree 2
2. Degree 4

a =1

a =1

𝒂≠𝟏

𝒂≠𝟏
Methods for ALL Degrees
1. GCF
2. Factor by Grouping
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Graphing Polynomials
1.




𝒇(𝒙) = 𝟐𝒙𝟑 − 𝟔𝒙𝟐
Degree = _________
EB: ___________
Poss. # of turns = _____________
x-intercepts =
Actual = ___________
 y-int = ______________
2.




𝑓(𝑥) = −𝑥 4 + 4𝑥 2
Degree = _________
EB: _____________
Poss. # of turns = _____________ Actual = __________
x-intercepts =

y-int = ______________
3. 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 𝑥 + 2
 Degree = _________
 EB: _____________
 Poss. # of turns = _____________ Actual = ___________
 x-intercepts =

y-int = _____________
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4.




𝑓(𝑥) = 𝑥 4 − 6𝑥 2 + 9
Degree = _________
EB: _____________
Poss. # of turns = _____________ Actual = __________
x-intercepts =

y-int = ______________
Method 1 – Long Division
Example 1: (𝑥 3 − 3𝑥 2 + 𝑥 − 8) ÷ (𝑥 − 1)
5. 𝑓(𝑥) = 2𝑥 5 + 4𝑥 4 − 3𝑥 3 − 6𝑥 2
 Degree = _________
 EB: _____________
 Poss. # of turns = ____________ Actual = __________
 x-intercepts =

y-int = ______________
Dividing Polynomials
Example 2: (−3𝑥 3 + 4𝑥 − 1) ÷ (𝑥 − 1)
Example 3: (3𝑥 3 − 2𝑥 2 + 4𝑥 − 1) ÷ (𝑥 2 + 1)
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Method 2 - Synthetic Division: process of dividing a polynomial by ______________________
Steps:
Example 1: Divide 𝑓(𝑥) = 𝑥 3 + 2𝑥 2 − 6𝑥 − 9 by 𝑥 + 3
1. Arrange the terms of the polynomial in
descending powers of x. Insert zeros for any
missing powers of x
2. Write the constant 𝑟 of the divisor 𝑥 − 𝑘.
3. Bring down the first coefficient.
4. Multiply the first coefficient by k. Then write the
product under the next coefficient. Add.
5. Multiply the sum by k. Then write the product
under the next coefficient. Add. Repeat Step 5 for
all coefficients in the dividend
6. Bottom line are the coefficients for each term.
Start one degree less than the dividend. Last
number is the remainder.
Example 2: Divide 𝑓(𝑥) = 𝑥 4 + 𝑥 3 − 5𝑥 2 + 𝑥 − 6 by 𝑥 − 2
The Remainder Theorem: If a polynomial 𝑓(𝑥) is divided by 𝑥 – 𝑘, the remainder (r) is the value of 𝑓(𝑘).
Ex. If 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 − 7𝑥 − 6 and 𝑓(5) = 159 then the remainder of
𝑥 3 +3𝑥 2 −7𝑥−6
𝑥−5
is 159.
Application: Determine the remainder when the polynomial 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 − 7𝑥 − 6 is divided by 𝑥 + 4.
The Factor Theorem: A polynomial 𝑓(𝑥) has a factor (𝑥 – 𝑘) if and only if 𝑓(𝑘) = 0.
Ex. Given 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 − 7𝑥 − 6 , since 𝑓(2) = 0 the remainder (after synthetic division) is zero therefore (𝑥 – 2)
is a factor and 𝑥 = 2 is a zero or root.
Application: Determine if the values are roots/zeros of the polynomial 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 9𝑥 + 18.
a. 𝑥 = 2
b. 𝑥 = −4
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Finding All Roots to a Polynomial
Rational Root Theorem
Example 1:
𝒇(𝒙) = 𝟐𝒙𝟒 + 𝒙𝟑 − 𝟏𝟒𝒙𝟐 − 𝟏𝟗𝒙 − 𝟔
1. List all possible p values (factors of p = constant)
2. List all possible q values (factors of q = leading coefficient)
3. List all possible roots
p
q
4. Test the roots using the remainder theorem.
5. Use synthetic division for the roots that work
6. Try to factor the depressed polynomial, or complete the
rational root test with the polynomial
Example 2: 𝑓(𝑥) = 2𝑥 4 − 9𝑥 2 + 7
Example 3: 𝑓(𝑥) = 3𝑥 4 − 10𝑥 3 − 24𝑥 2 − 6𝑥 + 5
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Analyzing Graphs of Polynomials:
Approximating a Relative Minimum and Relative Maximum:
1.
f ( x)  3 x 2  4 x  2
2.
f ( x)  x 3  3x 2  2
3.
Increasing and Decreasing Functions:
4.
f ( x)  x 3
5.
f ( x)  x 3  3x
6.
f ( x)  3x 4  6 x 2
t  1, t  0

f (t )  1, 0  t  2
t  3, t  2

Increasing:
Increasing:
Increasing:
Decreasing:
Decreasing:
Decreasing:
Calculator Work:
Sketch the graph of the following functions.
1. f (x )  2x 3  5x  6
2. f (x )  x 2  4x  1
Zeros:
Zeros:
Relative Maximums:
Relative Maximums:
Relative Minimums:
Relative Minimums:
Increasing:
Increasing:
Decreasing:
Decreasing:
Max/Min Calc Steps
Type graph in y =
Hit Zoom #6
Adjust Window to see a closer
view (if needed)
Hit Graph
Hit 2nd Calc #4 Max or #3 Min
Move cursor Left Bound, Enter
Move cursor Right Bound, Enter
Move cursor to the middle(Guess),
Enter
Zeros Steps
Type graph in y =
Hit Zoom #6
Adjust Window to see a closer
view (if needed)
Hit Graph
Hit 2nd Calc #2 (Zero)
Move cursor Left Bound, Enter
Move cursor Right Bound, Enter
Move cursor to the middle(Guess),
Enter
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