Polynomials R4

1
Polynomials (R4)
2
But first
Y
n
c
More about variables
X
x
X
m
x
Y
x
a
X
b
i
3
ax2 bx c y
We can parse the symbolic relationships thus
a x b x c y
2
x and y are input and output. But what about
a, b, and c?
4
Parameter Quantities
Input Output
a b c
x
y
The first letters of the alphabet are typically
used to represent parameters values that control
the transformation of input into output.
5
y a x
y 1 x y 2 x y 3 x y 4 x
The output is equal to the input.
The output is double the input.
The output is triple the input.
The output is quadruple the input.
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y a x
1
x
y
The are all similar, simply increasing the size
of the input. a is like the control knob,
changing the size of the output. The technical
term for these is coefficients, because they
together effect the result.
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x y
x x y
2
x y
Using parameters or co-efficients lets us talk
about similar types or families of
relationships.
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The Elementary School Polynomial
Can you say polynomial?
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1x3 2x2 4x1 3x0
What if x 10?
1(1000) 2(100) 4(10) 3(1) or just 1,243
10
1243
1 2 4 3
103 102 101 1
One thousand two hundred four tens three ones
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1 2 4 3
103 102 101 1
1 2 4 3
One thousand two hundred
four tens three ones
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Adding Polynomials
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8x3 2x2 6x 2
3x4 2x3 1x2 1x
3x4 10x3 3x2 7x 2
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8 2 6 2
x4 x3 x2 x1 x0
3 2 1 1
3 10 3 7 2
3x4 10x3 3x2 7x 2
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8x3 2x2 6x 2
3x4 - 2x3 - 1x2 1x
3x4 6x3 1x2 7x 2
Of course, be careful with the signs.
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8 2 6 2
x4 x3 x2 x1 x0
3 - 2 - 1 1
3 6 1 7 2
3x4 6x3 1x2 7x 2
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Multiplying Binomials
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F - O - I - L
Outer
Inner
(x 3) (x 3)
First
Last
x2 3x 3x 32
First Outer Inter
Last
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What is the area of each block?
3 x
x 3
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What is the area of each block?
3 x
x2
x 3
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What is the area of each block?
3 x
32
x2
x 3
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What is the area of each block?
3 x
3x
32
x2
3x
x 3
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What is the area of each block?
3x
32
x2
3x
x2 2(3x) 32
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What is the area of each block?
L
O
3 x
Outer
Last
F
I
Inner
First
x 3
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The Difference of
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(x a ) ( x a )
(10 1) (10 1) 9 11 99 102 12 (
2) ( 2) 8 12 96 102 - 22 (
3) ( 3) 7 13 91 102
32 ( 4) ( 4) 6 14 84 102
42 ( 5) ( 5) 5 15 75
102 - 52
(x2 a2 )
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Heres how
-2
x
x 2
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Heres how
x
x2
x
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Cut off and move the top
-2
x
x2
x
30
This is still x2.
x
x2
x 2
31
x2 take away 22
22
x2
32
x2 - 22
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Homework

• p. 42
• 27 84 (all)