01 polynomials, the building blocks of algebra college algebra
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01 Polynomials,The building blocks of algebra
College Algebra
Numbers
• Natural / Counting• Integers• Rational• Irrational
1.1 Underlying field of numbers
Real Numbers
Irrational
25
3 5 3
1.2 Indeterminates, variables, parameters
Given:
ax2 + bx + c
Usual thought:
x = variable
a, b, & c = constants
mx + c
you may recognize and associate this expression with a linear equation
The idea (and warning) is to
look for definitions
Likewise…
Linear equations
Most books teach the following:
• Slope Intercept Form:
• Standard Form:
• Point Slope Form:
y = mx + b
Ax + By = C
y – y1 = m(x - x1)
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
L
G
B
B2 + G2 = L2
Variables
• Used to represent and unknown quantity or a changing value.
x y + 2
3x – 2y 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 Bases
am * an
32 * 34
• Add exponents
am+n
32+4 = 36
Rules of Exponents: Exponents
• Exp raised to an Exp
(am )n
(32)4
• Multiply exponents
am*n
32*4= 38
Rules of Exponents: DIVISION
• Divide like Bases
am
an
34
32
• Subtract exponents
am-n
34-2 = 32
Rules of Exponents:
• Qty raised to an Exp
(ab)m
(3x)4
• Distribute exponents
ambm
34x4
Quantity to an Exponent
Rules of Exponents: Negative Exp
• Number raised to a neg Exp
a-m
3-2
• = the reciprocal
1
am
12 1
32 9=
Degrees of Polynomials
• 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
3x2y + 4xy
Degrees of Polynomials
• Generally speaking, the degree of 3x2y + 4xy = 3
• How will this change is y is defined as a constant and x is a variable?
3x2y + 4xy
Degrees of Polynomials
• 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
3x2y + 4xy
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 -2xx6-3x5+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 propertyThis 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)3-2(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
• add• subtract• multiply• divide• quantities• exponents
Be SPECIFIC!!!!!
• +• -• *• /• ( )• ^
Be SPECIFIC!!!!!
WebHomework Syntax
• 3x2y + 4xy
3*x^2*y+4*x*y• 4Ab - 5aB3
4*A*b-5*a*B^3 (Case Sensitive)
• Quantities
((7+x^2)/(2*z))*y• No extra spaces
yz
x
2
7 2
Free Mathematics Software
• http://math.exeter.edu/rparris/