Intermediate Algebra 098A Review of Exponents & Factoring 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.

n b 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 a a a m n

Quotient Rule m a m n a n

a Integer Exponent a n 1 n

a Zero as an exponent 0 a 1 a 0 R Calculator Key

^ Exponent Key Sample problem 3 0 8x y

2 5 24 x y 5 y 5 3x more exponents

Power to a Power n m a a mn

Product to a Power ab r r a b

r Polynomials - Review 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 8x 4 4 0.5x y

5 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 5 x x 10 x 9 x 1

2 3x 4 x 5 2 x 16 5 3

4 3 2 x 3 x y 2 xy y 6 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.

2. 3. 4. 5. 6. Plug In Use [Table] Use program EVALUATE Use [STO->]

Use [VARS] [Y=]] 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 b b b

r s 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

a b a b a 2 b 2 (a b) a 2 ab b 2

Squaring a Binomial 2 2 2 a b a 2ab b 2

2 2 a b a 2ab b Objective: Simplify Expressions Use techniques as part of a larger simplification problem.

Albert EinsteinPhysicist In the middle of difficulty lies opportunity. Intermediate Algebra 098A 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 Find the GCF of the terms 2. Factor each term with the

GCF as one factor. 3. Apply distributive property to factor the polynomial 1. Example of Common Factor 3 2

16 x y 40 x 2 8 x (2 xy 5) Factoring when first terms is negative Prefer the first term inside parentheses to be positive. Factor out the negative of the

GCF. 3 20 xy 36 y 2 4 y (5 xy 9) Factoring when GCF is a

polynomial a (c 5) b(c 5) (c 5)(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 grouping 2 2 32 xy 48 xy 20 y 30 y

2 y 16 xy 24 x 10 y 15 2 y 2 y 3 8 x 5 Ralph Waldo Emerson U.S. essayist, poet, philosopher We live in succession , in division, in parts, in

particles. Intermediate Algebra 098A 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 2

a b a b a b Special Note The sum of two squares is prime and cannot be factored. 2 2 a b is prime

Factoring Perfect Square Trinomials 2 2 2 2

2 2 a 2ab b a b a 2ab b a b Factor: Sum and Difference of cubes

a b (a b) a ab b 2 a b (a b) a ab b 2 3

3 3 3 2 2

Note The following is not factorable 2 a ab b

2 Factoring sum of Cubes informal (first + second) (first squared minus first times second plus second squared) Intermediate Algebra 098A Factoring Trinomials of

General Quadratic 2 ax bx c 50 y 15 y Objectives: Factor trinomials of the form

2 x bx c 2 ax bx c Factoring

2 x bx c 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 Factoring Trial and Error 2 ax bx c

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 Factoring ac method

2 ax bx c 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. Example of ac method 2 6 x 11x 4 2 6 x 3x 8 x 4 3x(2 x 1) 4(2 x 1)

(2 x 1)(3 x 4) Example of ac method 2 2 5 y (8 y 10 y 3) 5 y 8 y 2 y 12 y 3

2 2 2 5 y 2 y 4 y 1 3 4 y 1 2 5 y 4 y 1 2 y 3

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:

Ive always believed no matter how many shots I miss, Im going to make the next one. Intermediate Algebra 098A 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

x 5 x 2 0 x 5 0 or x 2 0 x 5 or x 2 5, 2 or 2, 5

Solving a polynomial equation by factoring. 1. 2. 3. 4. Factor the polynomial completely.

Set each factor equal to 0 Solve each of resulting equations Check solutions in original equation. 5. Write the equation in standard form. Example solve by factoring 2

3 x 11x 4 2 3 x 11x 4 0 3x 1 x 4 0 3 x 1 0 or x 4 0

1 x or x 4 3 Example: solve by factoring 3 2

3 2 x 4 x 12 x x 4 x 12 x 0 x x 4 x 12 0 2

x x 6 x 2 0 0, 6, 2 Example: solve by factoring 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 2 2 x x 1 x 9

2 x 20 or x 4 Answer: 20 ft, 21 ft, and 29 ft Example solve by factoring 3 x 2 x 7 12 Answer: {-1/2,4}

Example: solve by factoring 1 2 1 1 2 x 3 x x 2 2 12

3 Answer: {-5/2,2} Example: solve by factoring 9 y y 1 4 y 6 y 1 3 y 2 Answer: {0,4/3}

Example: solve by factoring 3 2 t 3t 13 7t 3t 1 Answer: {-3,-2,2} Sugar Ray Robinson

Ive always believed that you can think positive just as well as you can think negative. 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.