Download Chapter 5.2

Chapter 5.2
Evaluate & Graph
Polynomial Functions
#35 "In mathematics, you
don't understand things. You
just get used to them." -Johann von Neumann
Look at Polynomials, and how to name
them
 Evaluate by Synthetic Substitution
 And learn end behavior

Today we are going to…
Polynomials

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
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Term - Parts of an expression separated
by a (+) or (-) sign.
Monomial Expression w/ only one term.
Binomial Expression w/ two terms.
Trinomial Expression w/ three terms.
Polynomial - General name for
expressions with at least two terms.
Polynomials cannot have variables or
negative numbers for powers.
3x  4 x  7
2
Constant – Term w/o a variable
Leading Coefficient – The coefficient of
the
term
w/ the highest power.
 Degree of a Polynomial – The highest
power in a polynomial.
 Standard Form - Polynomials should
always be
written w/ the highest power
first and
descending to the lowest
power.
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
Parts of a Polynomial
How to name a polynomial

You can use substitution-from algebra one
◦ This is the plug in chug

Or you can use synthetic substitution, NEW
◦ Will become much quicker method than straight sub.
Pulse you will be forced to use it later on in the
chapter 
There are two ways to evaluate
Evaluate by Substitution
3
2
f ( x)  -4 x  5 x  7 x  6
when x  - 2
Replace x with -2 & simplify
f ( x)  - 4 x  5 x - 7 x  6
3
2
f ( x)  - 4(2)3  5(2)2 - 7(2)  6
f ( x)  - 4(8)  5(4) - 7( 2)  6
f ( x)  32  20  14  6
f ( x )  72
f ( x)  5x  2 x  8 x  26
3
2
when x  3
Replace x with 3 & simplify
You try to
Evaluate by
Substitution
f ( x)  5x  2 x -8 x  16
3
2
f ( x)  5(3)3  2(3)2 -8(3)  16
f ( x)  5(27)  2(9) - 8(3)  16
f ( x)  135-18-24  16
f ( x )  109
Evaluate by Synthetic Substitution
f ( x)  5x  2 x  8x  26 when x  3
3
3
3
3
2
0
-2
0
5
9
27 75
225
9
25
230
75
Evaluate by Synthetic Substitution
f ( x)  -4 x  5x  7 x  6 when x  - 2
3
1.
2.
3.
4.
5.
2
Label and Write all
coefficients including
any zeros inside the
box.
Write the x-value on the
outside of the box.
Bring down the leading
coefficient.
Multiply the leading
coefficient by the xvalue. Write this
number under the 2nd
coefficient.
Add these two numbers
& continue the process.
2
4
-4
5
-7
6
8
-26
66
13
-33
72
Example
f ( x)  3x  2 x  5 when x  3
1.
2.
3.
4.
5.
4
2
Write all coefficients
including any zeros
inside the box.
Write the x-value on the
outside of the box.
Bring down the leading 3
coefficient.
Multiply the leading
coefficient by the xvalue. Write this
number under the 2nd
coefficient.
Add these two numbers
& continue the process.
3
3
0
-2
0
5
9
27 75
225
9
25
230
75
End Behavior of Polynomials
Degree: Odd
 Leading Coeff:
Positive

1000
800
600
400
Function goes up to the
right and down to the
left.
-10
-8
200
0
-6
-4
-2
0
-200
-400
-600
-800
-1000
2
4
6
8
10
End Behavior of Polynomials
Degree: Odd
 Leading Coeff:
Negative

1500
1000
Function goes down
to the right and up
to the left.
-15
500
0
-10
-5
0
-500
-1000
5
10
15
End Behavior of Polynomials
Degree: Even
 Leading Coeff:
Positive

12000
10000
8000
Function goes up to
the right and up to
the left.
6000
4000
2000
0
-15
-10
-5
0
5
10
15
End Behavior of Polynomials
Degree: Even
 Leading Coeff:
Negative

2000
0
-15
Function goes down
to the right and
down to the left.
-10
-5
0
-2000
-4000
-6000
-8000
-10000
-12000
5
10
15
What's the end behavior?
12
11
3x  4 x  7
19
3x
 4x
19
 4x
93x
15
20
 3x  x  7
4

3x  x  87
4
9
9 x  47 x  13x  11x  78
7
4
 3x 
9
2
3
4 x   x  ex 
4
3
7
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p341
4-36 even
Assignment