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234833375
Name _______________________
Polynomial Functions
Vocabulary for Polynomial Functions:
Monomial = an expression that contains a real number, a variable or a product of both. It also has whole-number
exponents.
Polynomial = a monomial or a sum of monomials.
The degree of a term is the exponent of the variable in that term.
The coefficient of a term is the numeric multiplier of that term.
The standard form of a polynomial is when the terms are listed in descending order by degree.
The degree of a polynomial is the largest degree of any term of the polynomial.
Classifying a Polynomial: Write it in standard form and then…
By Degree:
Degree Name
0
constant
1
linear
2
quadratic
3
cubic
4
quartic
5
quintic
By # of Terms:
Degree Name
1
monomial
2
binomial
3
trinomial
4
4 term
polynomial
…
n-term
polynomial
Shape of Graphs:
Degree General Shape
0
horizontal line
1
line
2
U-shape
3
S-shape
4
U-shape or W-shape
5
S-shape with humps
The relative maximum value is the greatest y-value of
the points on an interval of the graph.
A relative minimum is the least y-value among nearby
points (on an interval) of the graph.
The x-intercepts of the graph of a function are called
zeros. They are the same as solutions to the equation
polynomial = 0 and are also called roots.
If a linear factor of a polynomial is repeated, then the zero is repeated. A repeated zero is called a multiple zero. A
multiple zero has a multiplicity equal to the number of times the zero occurs. Determines the shape.
y
f ( x)  0.1x  x  4   x  5 
3
2
has zeros at x  0, 4, and  5
x  0 multiple zero, multiplicity 3, looks cubic at x  0 .
x  4 multiple zero, multiplicity 2, looks quadratic at x  4 .
x  5 , is from a linear factor, looks like a line at x  5 .






x








Irrational Roots come in pairs
a  b and a  b called conjugates.
Imaginary (Complex) Roots come in pairs
a  bi and a  bi called complex conjugates.





S. Stirling
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
234833375
Name _______________________
Write Polynomial in Standard Form
(given factored form)
Multiply the factors together. (Each term of
one factor gets multiplied be each term
of the other factor.)
Write in highest degree to lowest degree.
y   x  1 x  1 x  2 mult. 2 factors
Write Polynomial in Factored Form (given
standard form)
Just keep factoring until you can’t anymore.
y  6 x3  15x 2  36 x factor out a GCF
y  3x  2 x 2  5 x  12  ac & b method on trinomial
y  3x  2 x 2  8x    3x  12 
y   x 2  1x  1x  1  x  2  now simplify
y  3x  2x  x  4   3  x  4  
y  x3  2 x 2  x  2 x 2  4 x  2 simplify
y  x3  4 x 2  5 x  2
Check by multiplication!
y   x 2  2 x  1  x  2  mult. 2 new factors
y  3x  2 x  3 x  4 final factored form.
Find the Polynomial from its Zeros
Write the linear factors from the zeros. It is
always (x – zero).
Simplify if possible.
Write in standard from (from factored form.)
Find the Zeros of a Polynomial
Get the equation = 0
Get it in factored form.
Set each factor = 0 and solve each equation.
Zeros of 3rd degree polynomial x = 2, –2, –1.
Set up factors from the zeros.
0  2 x 4  2 x3  24 x 2 mult. 2 factors
0  2 x 2  x 2  x  12  ac & b method
y   x  2 x  2 x  1 mult. 2 factors
y   x 2  2 x  2 x  4   x  1 now simplify
y   x 2  4   x  1 mult. 2 new factors
y  x3  1x 2  4 x  4 Write highest to lowest
degree.
S. Stirling
0  2 x 2  x 2  4 x    3x  12 
0  2 x 2  x  x  4   3  x  4  
0  2 x2  x  3 x  4 fully factored
Set each factor = 0 :
2 x 2  0 or x  3  0 or x  4  0 now solve
x  0 or x  3 or x  4
The zero of x  0 has multiplicity 2, so total 4.
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234833375
Name _______________________
Long Division
Multiply the quotient by divisor.
Subtract (remember to distribute the – )
Repeat until you get a remainder.
Divide x  3x  1 by x  4 .
Synthetic Division
Write the root in the “box” & all coefficients.
Bring the first number down.
Multiply root by number then add.
Repeat until you get a remainder.
2
x 1
x  4 x  3x  1
2
x3  5 x 2  4 x  20  x  5 .
5
1
 x2  4x
x 1
x4
5
Answer: x  1 R 5
1
5
4
20
5
0
20
0
4
0
Answer: 1x  0 x  4 or x  4
2
2
If the remainder is zero, you can factor
x3  5 x 2  4 x  20 into  x  5   x 2  4  .
Evaluate Using Synthetic Division
Write the x-value in the “box”
Do synthetic division.
The remainder is the y-value.
f ( x)  x3  5x 2  4 x  20 Find f (5) .
5
Calculator:
Enter the data into [STAT] EDIT
[STAT PLOT] turn it on.
[ZOOM]
Use [STAT] CALC
Choose the regression model that appears
to fit the data (by overall shape):
4: LinReg
5: QuadReg
6:CubicReg
7:QuartReg
Follow with L1, L2, Y1
Use [VARS] Y-VARS 1: Function
Choose the function and input the X
value. Y1(#)
S. Stirling
1
1
Answer:
5
4
20
5
0
20
0
4
0
f (5) = 0
Calculator Solutions [GRAPH]:
[Y=] enter left hand part Y1
enter left hand part Y2
[GRAPH]
[ZOOM] choose a window if necessary.
Look for all intersections
2nd [CALC] 5: intersect
Answer calculator’s questions.
Make sure you find all answers if there
are more than one point of intersection.
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