01 Polynomials

8/12/2019 01 Polynomials

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01 Polynomials,

The building blocks

of algebra

College Algebra



Active participation in

your education…

Timely analysis your own work

(personally I learn from my errors).

Verbalize situations.

What did I do wrong?

Why was it wrong?

What did I need to do right? Why isthis way right? Keep this in mind

as you take notes.



Numbers

• Natural /

Counting• Integers

• Rational• Irrational

1.1 Underlying field of numbers



Real Numbers

Irrational

25

35

3



1.2 Indeterminates, variables,

parameters

Given:

ax2 + bx + c

Usual thought:

x = variable

a, b, & c = constants





Textbooks (and some teachers)

lull us into complacency regarding

equations and format.

Linear equations is a good example ofthis mathematical rut.

(I hope to encourage your thinkingpatterns out of this mathematical rut

of how things are represented or

mentally translated.)



Linear equations

Most books teach the following:

• Slope Intercept Form:

• Standard Form:• Point Slope Form:

y = mx + b

Ax + By = Cy – 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





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:

Negative Exp

• Number

raised to a

neg Exp

a-m

3-2

• = the

reciprocal

1

a

m

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





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))





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?





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?





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

• 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

y

z

x

2

7 2



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01 Polynomials