Transcript 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
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.
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 3 5 x 2 - 7 x 6
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
when x 3
3
2
Replace x with 3 & simplify
You try to
Evaluate by
Substitution
f ( x) 5x3 2 x 2 -8x 16
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 ) 1 09
Evaluate by Synthetic Substitution
f ( x) 5x 2 x 8x 26 when x 3
3
3
2
3
0
9
-2
0
27 75
5
225
3
9
25 75
230
Evaluate by Synthetic Substitution
f ( x) -4 x 5 x 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
5
8
-7
-26
-4
13
-33
6
66
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
0
9
-2
0
27 75
5
225
3
9
25 75
230
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
p341
4-36 even
Assignment