Transcript 01 Polynomials

01 Polynomials,
The building blocks
of algebra
College Algebra
1.1 Underlying field of numbers
Numbers
• Natural /
Counting
• Integers
• Rational
• Irrational
Real Numbers
Irrational
3
5
5 2

3
1.2 Indeterminates, variables,
parameters
Given:
ax2 + bx + c
Usual thought:
x = variable
a, b, & c = constants
Likewise…
mx + c
you may recognize and associate
this expression with a linear
equation
The idea (and warning) is to
look for definitions
Linear equations
Most books teach the following:
• Slope Intercept Form: y = mx + b
Ax + By = C
• Standard Form:
y – y1 = m(x - x1)
• Point Slope Form:
These are the same types of
equations
• c = pn + d
• pn + c = d
• Profit = price*quanity - cost
Pythagorean Theorem is a good
example also a2 + b2 = c2
•
•
•
•
What if we are talking about a:
Building = B
Ladder = L
Ground Distance = G
B2 + G2 = L2
L
B
G
Variables
• Used to represent and
unknown quantity or a
changing value.
x
3x – 2y
y+2
mx + b
1.3 Basics of Polynomials
• Parts
– Coefficient
– Variable
– Terms
• Monomials
• Polynomial (multiple terms)
3x2y + 4xy
Remember you may have definitions
1.4 Working with
Polynomials
• To add or subtract one must
have like terms.
3xy + 4xy = 7xy
3xy+4x is in simplified form
Rules of Exponents:
MULTIPLICATION
• Multiply like • Add
exponents
Bases
m
a *
n
a
m+n
a
2
3
4
3
2+4
3
*
=
6
3
Rules of Exponents:
Exponents
• Exp raised
to an Exp
• Multiply
exponents
m
n
(a )
m*n
a
2
4
(3 )
2*4
3 =
8
3
Rules of Exponents:
DIVISION
• Divide like
Bases
• Subtract
exponents
am
an
m-n
a
34
32
4-2
3
=
2
3
Rules of Exponents:
Quantity to an
Exponent
• Qty raised
to an Exp
• Distribute
exponents
m
(ab)
m
m
a b
(3x)4
4
4
3x
Rules of Exponents:
Negative Exp
• Number
raised to a
neg Exp
a-m
• = the
reciprocal
3-2
12
2
3
1
am
1
=9
Degrees of Polynomials
3x2y + 4xy
• Degrees will be dependent on the
definition of the variables.
• The degree is the highest (combined
value) of the exponents of one term.
• Degree of x2y = 3
• Degree of xy = 2
Therefore the degree of 3x2y + 4xy = 3
Degrees of Polynomials
3x2y + 4xy
• Generally speaking, the degree of
3x2y + 4xy = 3
• How will this change is y is defined
as a constant and x is a variable?
Degrees of Polynomials
3x2y + 4xy
• Generally speaking, the degree of
3x2y + 4xy = 3
• How will this change is y is defined
as a constant and x is a variable?
• The Degree = 2 because 2 is the
highest exponent of the VARIABLE
1.5 Examples of Polynomial
Expressions
• What is the degree of f(x)?
f(x) = x6-3x5+3x4-2x3-2x2-x+3
• What is the degree?
11x4y-3x3y2+7x2y3-6xy4
• What is the degree if y is a
variable?
g(x) = 11x4y-3x3y3+7x2y3-2xy4
1.5 Examples of “NOW WHAT”
happens…Polynomial Expressions
f(x) = x6-3x5+3x4-2x3-2x2-x+3
g(x) = 11x4-3x3+7x2-2x
1. f(x)+g(x)
2. f(x)g(x)
3. f(g(x))
1.5 Examples of “NOW WHAT”
happens…Polynomial Expressions
f(x) = x6-3x5+3x4-2x3-2x2-x+3
g(x) = 11x4-3x3+7x2-2x
1. f(x)+g(x)
x6-3x5+3x4-2x3 -2x2 -x+3
+
11x4-3x3+7x2 -2x
x6-3x5+14x4-5x3+5x2-3x+3
Possible questions..
What is the degree? What is the
coefficient of the x cubed term?
1.5 Examples of “NOW WHAT”
happens…Polynomial Expressions
f(x) = x6-3x5+3x4-2x3-2x2-x+3
g(x) = 11x4-3x3+7x2-2x
2. f(x)g(x) -- distributive property
This could be ugly if one was asked to complete the multiplication
(x6-3x5+3x4-2x3-2x2-x+3)(11x4-3x3+7x2-2x)=
11x10-3x9+7x8 -2x7
-33x9+9x8-21x7+6x6
+33x8-9x7+21x6-6x5
… what is the degree of the product?
1.5 Examples of “NOW WHAT”
happens…Polynomial Expressions
f(x) = x6-3x5+3x4-2x3-2x2-x+3
g(x) = (11x4-3x3+7x2-2x)
3. f(g(x))
(11x4-3x3+7x2-2x)6-3(11x4-3x3+7x2-2x)5
+3(11x4-3x3+7x2-2x)4-2(11x4-3x3+7x2-2x)32(11x4-3x3+7x2-2x)2-(11x4-3x3+7x2-2x)+3 =
(11x4-3x3+7x2-2x)6- …
116x24-36x18+76x12-64x6- …
what is the degree?
WebHomework Syntax
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add
subtract
multiply
divide
quantities
exponents
Be SPECIFIC!!!!!
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+
*
/
( )
^
Be SPECIFIC!!!!!
WebHomework Syntax
• 3x2y + 4xy
3*x^2*y+4*x*y
• 4Ab - 5aB3
4*A*b-5*a*B^3 (Case
Sensitive)
2
7

x
• Quantities
y
2z
((7+x^2)/(2*z))*y
• No extra spaces
Free Mathematics Software
• http://math.exeter.edu/rparris/