Intermediate Algebra Chapter 5 – Martin Gay Polynomials.

Intermediate AlgebraChapter 5 – Martin Gay

•Polynomials

1.1 – Integer Exponents

• For any real number b and any natural number n, the nth power of b is found by multiplying b as a factor n times.

nb b b b b

Exponential Expression – an expression that involves

exponents

• Base – the number being multiplied

• Exponent – the number of factors of the base.

Product Rule

n n m na a a

Quotient Rule

mm n

n

aa

a

Integer Exponent

1nn

aa

Zero as an exponent

0 1 0a a R

Calculator Key

• Exponent Key

^

Sample problem

3 0

2 5

8

24

x y

x y

5

53

y

x

Section 5.2 – more exponents

• Power to a Power

nm mna a

Product to a Power

r r rab a b

Intermediate Algebra – 5.3

•Addition

•and

•Subtraction

Objective:

•Determine the coefficient and degree of a monomial

Def: Monomial

• An expression that is a constant or a product of a constant and variables that are raised to whole –number powers.

• Ex: 4x 1.6 2xyz

Definitions:

• Coefficient: The numerical factor in a monomial

• Degree of a Monomial: The sum of the exponents of all variables in the monomial.

Examples – identify the degree

48x4 50.5x y

4

5

Def: Polynomial:

•A monomial or an expression that can be written as a sum or monomials.

Def: Polynomial in one variable:

•A polynomial in which every variable term has the same variable.

Definitions:

• Binomial: A polynomial containing two terms.

• Trinomial: A polynomial containing three terms.

Degree of a Polynomial

•The greatest degree of any of the terms in the polynomial.

Examples:

6 3 2

2

2

5 3 4 3 2

5 10 9 1

3 4 5

16

3 2 6

x x x x

x x

x

x x y xy y

Objective

•Add •and

•Subtract •Polynomials

To add or subtract Polynomials

• Combine Like Terms

• May be done with columns or horizontally

• When subtracting- change the sign and add

Evaluate Polynomial Functions

• Use functional notation to give a polynomial a name such as p or q and use functional notation such as p(x)

• Can use Calculator

Calculator Methods

• 1. Plug In

• 2. Use [Table]

• 3. Use program EVALUATE

• 4. Use [STO->]

• 5. Use [VARS] [Y=]

• 6. Use graph- [CAL][Value]

Objective:

•Apply evaluation of polynomials to real-life applications.

Intermediate Algebra 5.4

•Multiplication

•and

•Special Products

Objective

• Multiply • a

• polynomial • by a

• monomial

Procedure: Multiply a polynomial by a monomial

• Use the distributive property to multiply each term in the polynomial by the monomial.

• Helpful to multiply the coefficients first, then the variables in alphabetical order.

Law of Exponents

r s r sb b b

Objectives:

• Multiply Polynomials

• Multiply Binomials.• Multiply Special

Products.

Procedure: Multiplying Polynomials

• 1. Multiply every term in the first polynomial by every term in the second polynomial.

• 2. Combine like terms.• 3. Can be done horizontally or

vertically.

Multiplying Binomials

• FOIL• First• Outer• Inner• Last

Product of the sum and difference of the same two terms

Also called multiplying conjugates

2 2a b a b a b

Squaring a Binomial

2 2 2

2 2 2

2

2

a b a ab b

a b a ab b

3 3

3 3 2 2

( )

( )

a b a b

a b a b a ab b

Objective:

• Simplify Expressions

• Use techniques as part of a larger simplification problem.

Albert Einstein-Physicist

•“In the middle of difficulty lies opportunity.”


•Common Factors

•and

•Grouping

Def: Factored Form

•A number or expression written as a product of factors.

Greatest Common Factor (GCF)

• Of two numbers a and b is the largest integer that is a factor of both a and b.

Calculator and gcd

• [MATH][NUM]gcd(

• Can do two numbers – input with commas and ).

• Example: gcd(36,48)=12

Greatest Common Factor (GCF) of a set of terms

•Always do this FIRST!

Procedure: Determine greatest common factor GCF of 2 or more monomials

• 1. Determine GCF of numerical coefficients.

• 2. Determine the smallest exponent of each exponential factor whose base is common to the monomials. Write base with that exponent.

• 3. Product of 1 and 2 is GCF

Factoring Common Factor

• 1. Find the GCF of the terms

• 2. Factor each term with the GCF as one factor.

• 3. Apply distributive property to factor the polynomial

Example of Common Factor

3 2

2

16 40

8 (2 5)

x y x

x xy

Factoring when first terms is negative

• Prefer the first term inside parentheses to be positive. Factor out the negative of the GCF.

3

2

20 36

4 (5 9)

xy y

y xy

Factoring when GCF is a polynomial

( 5) ( 5)

( 5)( )

a c b c

c a b

Factoring by Grouping – 4 terms

• 1. Check for a common factor

• 2. Group the terms so each group has a common factor.

• 3. Factor out the GCF in each group.

• 4. Factor out the common binomial factor – if none , rearrange polynomial

• 5. Check

Example – factor by grouping2 232 48 20 30xy xy y y

2 16 24 10 15y xy x y

2 2 3 8 5y y x

Ralph Waldo Emerson – U.S. essayist, poet, philosopher

•“We live in succession , in division, in parts, in particles.”


•Special Factoring

Objectives:Factor

• a difference of squares

• a perfect square trinomial

• a sum of cubes

• a difference of cubes

Factor the Difference of two squares

2 2a b a b a b

Special Note

• The sum of two squares is prime and cannot be factored.

2 2a b

is prime

Factoring Perfect Square Trinomials

22 2

22 2

2

2

a ab b a b

a ab b a b

Factor: Sum and Difference of cubes

3 3 2 2

3 3 2 2

( )

( )

a b a b a ab b

a b a b a ab b

Note

• The following is not factorable

2 2a ab b

Factoring sum of Cubes - informal

• (first + second)

• (first squared minus first times second plus second squared)


• Factoring Trinomials

• of

• General Quadratic

2ax bx c

Objectives:

• Factor trinomials of the form

4 3 240 50 15y y y

2

2

x bx c

ax bx c

Factoring

• 1. Find two numbers with a product equal to c and a sum equal to b.

• The factored trinomial will have the form(x + ___ ) (x + ___ )

• Where the second terms are the numbers found in step 1.

• Factors could be combinations of positive or negative

2x bx c

FactoringTrial and Error • 1. Look for a common factor

• 2. Determine a pair of coefficients of first terms whose product is a

• 3. Determine a pair of last terms whose product is c

• 4. Verify that the sum of factors yields b

• 5. Check with FOIL Redo

2ax bx c

Factoring ac method

• 1. Determine common factor if any

• 2. Find two factors of ac whose sum is b

• 3. Write a 4-term polynomial in which by is written as the sum of two like terms whose coefficients are two factors determined.

• 4. Factor by grouping.

2ax bx c

Example of ac method

26 11 4x x 26 3 8 4x x x

3 (2 1) 4(2 1)x x x

(2 1)(3 4)x x

Example of ac method

2 25 (8 10 3)y y y

2 25 8 2 12 3y y y y

25 2 4 1 3 4 1y y y y

25 4 1 2 3y y y

Factoring - overview• 1. Common Factor

• 2. 4 terms – factor by grouping

• 3. 3 terms – possible perfect square

• 4. 2 terms –difference of squares

• Sum of cubes

• Difference of cubes

• Check each term to see if completely factored

Isiah Thomas:

• “I’ve always believed no matter how many shots I miss, I’m going to make the next one.”


•Solving Equations

•by

•Factoring

Zero-Factor Theorem

•If a and b are real numbers

•and ab =0

•Then a = 0 or b = 0

Example of zero factor property

5 2 0

5 0 2 0

5 2

5,2 2, 5

x x

x or x

x or x

or

Solving a polynomial equation by factoring.

1. Factor the polynomial completely.

2. Set each factor equal to 03. Solve each of resulting equations4. Check solutions in original

equation.5. Write the equation in standard

form.

Example – solve by factoring

23 11 4x x 23 11 4 0x x

3 1 4 0x x 3 1 0 4 0x or x

14

3x or x

Example: solve by factoring

3 2

3 2

2

4 12

4 12 0

4 12 0

6 2 0

0,6, 2

x x x

x x x

x x x

x x x


• A right triangle has a hypotenuse 9 ft longer than the base and another side 1 foot longer than the base. How long are the sides?

• Hint: Draw a picture• Use the Pythagorean theorem

Solution

• Answer: 20 ft, 21 ft, and 29 ft

2 22 1 9x x x

20 4x or x

Example – solve by factoring

• Answer: {-1/2,4}

3 2 7 12x x


• Answer: {-5/2,2}

2 21 1 13 2

2 12 3x x x


• Answer: {0,4/3}

29 1 4 6 1 3y y y y y


• Answer: {-3,-2,2}

3 23 13 7 3 1t t t t

Sugar Ray Robinson

• “I’ve always believed that you can think positive just as well as you can think negative.”


•Division

Long division Problems

2 5 7

2

x x

x

Long Division Problem 2

4 29 5 7 10

3 1

x x x

x

Ans to long division problem 2

3 2

3 2

93 2

3 19

3 23 1

x x xx

x x xx

Long division Problems

2 5 7

2

x x

x

Maya Angelou - poet

• “Since time is the one immaterial object which we cannot influence – neither speed up nor slow down, add to nor diminish – it is an imponderably valuable gift.”

Intermediate Algebra Chapter 5 – Martin Gay Polynomials.

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