01 Polynomials, The building blocks of algebra College Algebra.

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/