Name: Directions: Use pencil and the space provided next to the question to
|
|
- Damon Bryan Gordon
- 5 years ago
- Views:
Transcription
1 Name: Directions: Use pencil and the space provided next to the question to show all work. The purpose of this packet is to give you a review of basic skills. Please refrain from using a calculator! Prepared by: Mrs. Trebat Fairfield Ludlowe High School mtrebat@fairfield.k1.ct.us 1
2 BRUSHING UP ON BASIC ALGEBRA SKILLS Mrs. Trebat Name: DUE DATE: Directions: Use pencil, show work, box in your answers. Monomial Factors of Polynomials A monomial is an expression that is either a numeral, a variable, or the product of a 3 numeral and one or more variables. Example of monomials: 7, x, 6x y. A sum of monomials is called a polynomial. Some polynomials have special names: Binomials (two terms): 3x 5 or xy x Trinomials (3 terms): x 5x 15 or x 6xy 9y Divide: 4x 1 1) 6 ) 4 3 8x 4x 6x x Factor (monomials only): Example: 15a 5b 35 5(3a 5b 7) 3) a a a 4) 3 5ax 10a x 15a Simplify: Example: 14x 1 10x 5 7(x 3) 5(x 5) x 3 x Factor 5) 6 a 9 b 7 a 1 b = 3 7 Cancel out common factors Simplify and combine 6) x y 3x y 6xy 9xy xy 3y
3 Multiplying Binomials Mentally Write each product as a trinomial: 7) ( x 9)( x 4) 8) (4 x)(1 x) = 9) (a 5)( a ) 10) (x 5)(3 x 4) Find the values of p, q, and r that make the equation true. 11) ( )( 5) 6 11 px q x x x r Difference of Two Squares You must use the shortcut below (do not FOIL!!!!) ( a b)( a b) a b ( First Second ) ( First Second ) First Second Write each product as a binomial: 1) ( x 7)( x 7) 13) ( y 8)( y 8) 14) (5x )(5x ) = 15) 8x 118x 11 16) 4a 5b 4a 5b 17) x 9y x 9y Now let s try reversing the process above Factor: 18) x 36 19) m 81 0) 5a 1 = 1) 49x 9y Factor each expression as the difference of two squares. Then simplify. Example: x x 3 x x 3 x x 3 3x 3 ) Apply the formula x 4 x 3) simplify 9( x 1) 4( 1) x = 3
4 Squares of Binomials and Perfect Square Trinomials Every time you square a binomial, the same pattern comes up. To speed up the process, we should memorize: a b a ab b and a b a ab b A trinomials is called a perfect square trinomial if it is the square of a binomial. For example, x 6x 9 is a perfect square trinomial because it is equal to x 3. Write each square as a trinomial. 4) a 9 = 5) x 7 6) 4x 1 7) 5a b Decide if each trinomial is a perfect square. If it is, factor it. Otherwise, write not a perfect square. 8) a 6a 9 9) y 14y 49 30) 11 x x 31) 9a 30ab 100b 3) 49x 8xy 4y 33) 5x 15xy 36y = 34) a b 1ab 36 35) x 9x 36) Show that a 8a 16 can be factored as ( a ) a. 4 37) Solve and check: x x
5 Factoring Quadratic Trinomials To factor a trinomial of the form x bx c, you must find two numbers, r and s, whose product is c and whose sum is b. x bx c x r x s When you find the product x r x s you obtain x bx c x ( r s ) x rs Example: Factor x x 15. a. List the factors of -15 (the last term). Factors Sum of the b. Either write them down or do this mentally. of -15 factors Find the pairs of factors with sum - (the middle term). 1, (discard) -3, 5 (discard) 3, -5 - (keep) x x 15 ( x 5)( x 3) Check the result by multiplying Factor. Check by multiplying (mentally). 38. x 8x x 7x x 4x x 9x x 3x x 11x x 5x x 15x x 9x x 3x 8 5
6 Factoring General Quadratic Trinomials of the type ax bx c Example: Factor = (x 9)( x 1) x 7x 9 Factor: List all factors List factors of -9 Of x 48) 3x 7x 49) x 5x 3 50) x 15x 7 51) 3a 4a 4 5) 5a 6a 53) 3x x 5 54) 3m 7m 6 55) 4a a 3 Factor by Grouping Example 1: 7( a ) 3 a( a ) (7 3 a)( a ) Notice that we factored out the common factor (a - ) Example : 5( x 3) x(3 x) Notice that x 3 and 3 x are opposites. 5( x 3) x(3 x) 5( x 3) x( x 3) (5 x)( x 3) 56) x ( x y ) y ( y x ) 57) 3 a( b a) b( a b) Group and Factor: 58) 3a ab 3c bc 59) 3 3a a 6a 60) 3ab b 4 1a 6
7 Solving Equations by Factoring The Zero Product Property Key Concept: Zero-Product Property For all real numbers a and b: a b 0 if and only if a 0 or b 0 A product of factors is equal to zero if and only if at least one of the factors is 0. Solve: 61) ( x 5)( x 3) 0 6) xx ( 9) 0 63) (a 3)(3 a ) 0 64) 3 x(5x )( x 7) 0 65) a 3a 0 66) x 1x ) b 4b 3 68) 5x ) 7x 18x 11 70) 3 8y y 0 71) 3 4x 1x 8x 0 7) 3 9x 5x 30x Sample for items 73-74: ( a 1)( a 3) 1 a a expand the left side; bring over 1; set it = 0; ( a 3)( a 5) 0 combine like terms; factor it; a 3or a 5 73) ( x 1)( x 5) 16 74) (z 5)( z 1) 7
8 Simplifying Fractions Follow the examples below. Example 1: Simplify 3 x 6. 3x 3y 3x 6 3( x ) Solution: 3x 3y 3( x y ) x, x y x y Factor the numerator and denominator; look for common factors; Cancel out common factor which is 3; x 9 Example : Simplify (x 1)(3 x) Solution: x 9 ( x 3)( x 3) (x 1)(3 x ) (x 1)( x 3) x 3, x 1, x 3 x 1 First factor the numerator; pull out a negative in the factor (3-x) to make it a common factor with the numerator; exclude the first as it would make the denominator =0; exclude the nd as it would make both numerator and denominator =0. Simplify. Give any restrictions on the variable. 75) 5 x 10 x a 4 76) a 4 x 77) 5 7 x (5 x)(7x ) 78) x 8x x 8
9 Multiplying Fractions Multiplication Rule for Fractions a c ac To multiply fractions, you multiply their numerators and multiply their denominators. b d bd Note: You can multiply first and then simplify, or you can simplify first and then multiply. Example: x x 1 x 5 x 4 x 3 x 5 x 5 x 5x x 3 x x 5 x 3 = x 4 x 5 x Notice how common factors were cancelled. Simplify. a 3a 79) a a 4 80) a x a a 3x 3a 81) A triangle has base 3 x 8 cm and height cm. What is its area? 4 9x Dividing Fractions a c a d Division Rule for Fractions: Example : b d b c To divide by a fraction, you multiply by its reciprocal! 9
10 8) a 1 a 83) 6 9 x 1 x ) a b a b 85) a 4a 4x 5 1x 30 x 16 x 8x Adding and Subtracting Algebraic Fractions Key Steps: (1) Find the Least Common Denominator (LCD); () Re-write each fraction being added or subtracted with the same common denominator. (3) Add or subtract their numerators and write the result over the common denominator. Example: x 30 9x 45 6( x 5) 9( x 5) ( x 5) 18( x 5) 5 5 = or 18( x 5) 18x 90 86) 4 x 3 7 x x Factor out denominators to more easily identify LCD multiply the first fraction by 3, the second by 10
11 87) 4 x 3 x 3 = x 4 88) x 1 x 1 89) 3a 6b a b b a x 11 x 7 90) = x 9 x 3x 11
12 BRUSHING UP ON BASIC ALGEBRA SKIUS - '501ue4 Mn. Treht Name:.- birections: Use pencil: You must show work for credit. Your fim/ansners must be clear& identified Monomial Factors of Polynomials A monomial is an expression that is either a numeral, a variable, or the product of a numeral and one or more variables. Example of monomials: 7, x, 6xy3. A sum of monomials is called a polynomial. Some polynomials have speaal names: Binomials (two terms): 3x - 5 or xy + x Trinomials (3 terms): xt + 5x - 15 or xt - 6xy + 9y Divide: Multiptyinq Binomials Mentally Write each product as a trinomial: 7)(x-9)(~+4)= xz-sx-36 8)(4-x)(1-x)= L f - 5 ~ t n ~ w Find the values of p, q, and r that make the equation true. 11) (px + q)(x + 5) = 6x + 11x + r apx+ (cp+nq>r +5q = brztlln tr dp?+spx +qx+ 57 = ~~uafi'ny I'-lc= rerrr': bifference of Two Squares p z,a ~ 5?+p=li 5(&)=,y+Zqzll \-IO-r Write each product as a binomial: 1) (x+7)(x-7) = x- yq 13) (y+8)(y-8)= y-67 Factor: Example: 15a = 5(3a ) - Simplify: Example: Now let's try reversing the process above... Factor: 5) Factor Cancel out common factors Simplify ond combine 6a+9b 7a+lb- j/(zai3b) - j (e+3b), at3b-a-3b=a 3 7 P f Factor each expression as the difference of two squares. Then simplify. Apply the formula simplify
13 Sauares of Binomials and Perfect Square Trinomials Every time you square a binomial, the same pattern comes up. To speed up the A trinomials is called a perfect square trinomial if it is the square of a binomial. For example, x - 6 x + 9 is a perfect square trinomial because it is equal to (x - 3y. Write euch square as a trinomial. 4) (a- 9) = az- I~L + d I 5) (x+7)= y+1lllc+ll(l Factorinq Quadratic Trinomials To factor a trinomial of the form x + bx + c, you must find two numbers, r and s, whose product is c and whose sum is b. x+bx+c=(x+r)(x+s) When you find the product (x + r)(x + s) you obtain Example: Factor x - x a. List the factors of -15 (the last term). Factors Sumof the b. Either write them down or do this mentally. of -15 factors Find the pairs of factors with sum - (the middle term). *discard) 1, (discard) Decide if each trinomial is a perfect square. If it is, factor it. Otherwise, write not a perfect square. 8) az + 6a + 9 = (a+ 3)' 9) y-14y+49= t 30) 11-x+x =_ ([I-K) 31) 9a + 30ab + 100bz = hlot A PtwkzrSQ+~~fc ( due +D middle Tunl.,, ) :.xz-x-15=(x-5)(x+3) Check the result by multiplying... Factor. Check by multiplying (mentally). 3, -5 - (keep) 36) Show that a4-8az + 16 can be factored as (a + )'(a -). a'- ~ o - ~ t l b = (az-'1)' = C&+~)(a-r)]~ = (G+~~(G-~I~ 37) Solve and check: (x + ) -(x- 3)t = 35 - (4-i-br +q) =35- qr+~+dr-~i = 33- LOX -5 = 3s 1 0 = ~ (lo
14 m &La C + M O O L h rn
15 Simplifiinq Fractions - Follow the examples below. 3x + 6 / Example 1: Simplify - i Solution: 3x+3y' 3x -- 3(x 3x + 3y 3(x + y), Factor the numerator and demmimtor look for common factors: 1 x + j --, X # -y Cnncel out common factor which is 3; x + Y xz-9 Example : Simplify (x + 1)(3 - x) Solution: xz-9 = (x + 3)(x - 3) First factor the numerutor; 'pull out' a (zx + 1)(3 - x) -(zx + l)(x - 3) "Wve in the fattor (3-X) to mk it a common factor with the numerator: x + 3 exclude the first m it would make the denomimator =O; uci& the '"'' it would mke both numerator and denomi~tor =O. I b Multiplyinq Fractions 1 Multiplication Rule for Fractions I -.-=a c ac d bd To multiply fractions, you multiply their numerators and multiply their denominators. Note: You can multiply first and then simplify, or you can simplify first and then multiply. Example: x - x - 1.-= xz - 5 (x - 4)(x/) (&)(x + 5) xz-5x x+3 Simplify. (x -4)(x + 5) Notice how common factors were cancelled. X Simplify. Give any restrictions on the variable. 3x 8 81) A triangle has base -cm 4 and height -cm. 9x What is its area?,, & ~&wc)[he~hi) L Dividi~ Fractions
16
-5y 4 10y 3 7y 2 y 5: where y = -3-5(-3) 4 10(-3) 3 7(-3) 2 (-3) 5: Simplify -5(81) 10(-27) 7(9) (-3) 5: Evaluate = -200
Polynomials: Objective Evaluate, add, subtract, multiply, and divide polynomials Definition: A Term is numbers or a product of numbers and/or variables. For example, 5x, 2y 2, -8, ab 4 c 2, etc. are all
More informationPolynomial and Rational Expressions. College Algebra
Polynomial and Rational Expressions College Algebra Polynomials A polynomial is an expression that can be written in the form a " x " + + a & x & + a ' x + a ( Each real number a i is called a coefficient.
More informationMultiplication of Polynomials
Multiplication of Polynomials In multiplying polynomials, we need to consider the following cases: Case 1: Monomial times Polynomial In this case, you can use the distributive property and laws of exponents
More informationName. 5. Simplify. a) (6x)(2x 2 ) b) (5pq 2 )( 4p 2 q 2 ) c) (3ab)( 2ab 2 )(2a 3 ) d) ( 6x 2 yz)( 5y 3 z)
3.1 Polynomials MATHPOWER TM 10, Ontario Edition, pp. 128 133 To add polynomials, collect like terms. To subtract a polynomial, add its opposite. To multiply monomials, multiply the numerical coefficients.
More information7.1 Review for Mastery
7.1 Review for Mastery Factors and Greatest Common Factors A prime number has exactly two factors, itself and 1. The number 1 is not a prime number. To write the prime factorization of a number, factor
More informationMATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: , , and 10.1)
NOTE: In addition to the problems below, please study the handout Exercise Set 10.1 posted at http://www.austin.cc.tx.us/jbickham/handouts. 1. Simplify: 5 7 5. Simplify: ( 6ab 5 c )( a c 5 ). Simplify:
More informationQuadratic Algebra Lesson #2
Quadratic Algebra Lesson # Factorisation Of Quadratic Expressions Many of the previous expansions have resulted in expressions of the form ax + bx + c. Examples: x + 5x+6 4x 9 9x + 6x + 1 These are known
More informationPOD. Combine these like terms: 1) 3x 2 4x + 5x x 7x ) 7y 2 + 2y y + 5y 2. 3) 5x 4 + 2x x 7x 4 + 3x x
POD Combine these like terms: 1) 3x 2 4x + 5x 2 6 + 9x 7x 2 + 2 2) 7y 2 + 2y 3 + 2 4y + 5y 2 3) 5x 4 + 2x 5 5 10x 7x 4 + 3x 5 12 + 2x 1 Definitions! Monomial: a single term ex: 4x Binomial: two terms separated
More informationAlg2A Factoring and Equations Review Packet
1 Factoring using GCF: Take the greatest common factor (GCF) for the numerical coefficient. When choosing the GCF for the variables, if all the terms have a common variable, take the one with the lowest
More informationUnit 8 Notes: Solving Quadratics by Factoring Alg 1
Unit 8 Notes: Solving Quadratics by Factoring Alg 1 Name Period Day Date Assignment (Due the next class meeting) Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday
More informationHow can we factor polynomials?
How can we factor polynomials? Factoring refers to writing something as a product. Factoring completely means that all of the factors are relatively prime (they have a GCF of 1). Methods of factoring:
More informationSlide 1 / 128. Polynomials
Slide 1 / 128 Polynomials Slide 2 / 128 Table of Contents Factors and GCF Factoring out GCF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring
More informationMATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: , , and 10.1)
NOTE: In addition to the problems below, please study the handout Exercise Set 10.1 posted at http://www.austincc.edu/jbickham/handouts. 1. Simplify: 5 7 5. Simplify: ( ab 5 c )( a c 5 ). Simplify: 4x
More informationUnit 8: Polynomials Chapter Test. Part 1: Identify each of the following as: Monomial, binomial, or trinomial. Then give the degree of each.
Unit 8: Polynomials Chapter Test Part 1: Identify each of the following as: Monomial, binomial, or trinomial. Then give the degree of each. 1. 9x 2 2 2. 3 3. 2x 2 + 3x + 1 4. 9y -1 Part 2: Simplify each
More informationALGEBRAIC EXPRESSIONS AND IDENTITIES
9 ALGEBRAIC EXPRESSIONS AND IDENTITIES Exercise 9.1 Q.1. Identify the terms, their coefficients for each of the following expressions. (i) 5xyz 3zy (ii) 1 + x + x (iii) 4x y 4x y z + z (iv) 3 pq + qr rp
More informationDevelopmental Math An Open Program Unit 12 Factoring First Edition
Developmental Math An Open Program Unit 12 Factoring First Edition Lesson 1 Introduction to Factoring TOPICS 12.1.1 Greatest Common Factor 1 Find the greatest common factor (GCF) of monomials. 2 Factor
More informationACCUPLACER Elementary Algebra Assessment Preparation Guide
ACCUPLACER Elementary Algebra Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre
More informationCCAC ELEMENTARY ALGEBRA
CCAC ELEMENTARY ALGEBRA Sample Questions TOPICS TO STUDY: Evaluate expressions Add, subtract, multiply, and divide polynomials Add, subtract, multiply, and divide rational expressions Factor two and three
More informationMultiplying Polynomials
14 Multiplying Polynomials This chapter will present problems for you to solve in the multiplication of polynomials. Specifically, you will practice solving problems multiplying a monomial (one term) and
More informationPolynomials * OpenStax
OpenStax-CNX module: m51246 1 Polynomials * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section students will: Abstract Identify
More informationWe begin, however, with the concept of prime factorization. Example: Determine the prime factorization of 12.
Chapter 3: Factors and Products 3.1 Factors and Multiples of Whole Numbers In this chapter we will look at the topic of factors and products. In previous years, we examined these with only numbers, whereas
More informationa*(variable) 2 + b*(variable) + c
CH. 8. Factoring polynomials of the form: a*(variable) + b*(variable) + c Factor: 6x + 11x + 4 STEP 1: Is there a GCF of all terms? NO STEP : How many terms are there? Is it of degree? YES * Is it in the
More informationAlg2A Factoring and Equations Review Packet
1 Multiplying binomials: We have a special way of remembering how to multiply binomials called FOIL: F: first x x = x 2 (x + 7)(x + 5) O: outer x 5 = 5x I: inner 7 x = 7x x 2 + 5x +7x + 35 (then simplify)
More informationSection 5.3 Practice Exercises Vocabulary and Key Concepts
Section 5.3 Practice Exercises Vocabulary and Key Concepts 1. a. To multiply 2(4x 5), apply the property. b. The conjugate of 4x + 7 is. c. When two conjugates are multiplied the resulting binomial is
More informationDownloaded from
9. Algebraic Expressions and Identities Q 1 Using identity (x - a) (x + a) = x 2 a 2 find 6 2 5 2. Q 2 Find the product of (7x 4y) and (3x - 7y). Q 3 Using suitable identity find (a + 3)(a + 2). Q 4 Using
More informationAlgebra. Chapter 8: Factoring Polynomials. Name: Teacher: Pd:
Algebra Chapter 8: Factoring Polynomials Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Factor polynomials by using the GCF. Pgs: 1-6 HW: Pages 7-8 o Day 2: SWBAT: Factor quadratic trinomials of
More information2 TERMS 3 TERMS 4 TERMS (Must be in one of the following forms (Diamond, Slide & Divide, (Grouping)
3.3 Notes Factoring Factoring Always look for a Greatest Common Factor FIRST!!! 2 TERMS 3 TERMS 4 TERMS (Must be in one of the following forms (Diamond, Slide & Divide, (Grouping) to factor with two terms)
More informationMATH 181-Quadratic Equations (7 )
MATH 181-Quadratic Equations (7 ) 7.1 Solving a Quadratic Equation by Factoring I. Factoring Terms with Common Factors (Find the greatest common factor) a. 16 1x 4x = 4( 4 3x x ) 3 b. 14x y 35x y = 3 c.
More informationTERMINOLOGY 4.1. READING ASSIGNMENT 4.2 Sections 5.4, 6.1 through 6.5. Binomial. Factor (verb) GCF. Monomial. Polynomial.
Section 4. Factoring Polynomials TERMINOLOGY 4.1 Prerequisite Terms: Binomial Factor (verb) GCF Monomial Polynomial Trinomial READING ASSIGNMENT 4. Sections 5.4, 6.1 through 6.5 160 READING AND SELF-DISCOVERY
More informationFACTORING HANDOUT. A General Factoring Strategy
This Factoring Packet was made possible by a GRCC Faculty Excellence grant by Neesha Patel and Adrienne Palmer. FACTORING HANDOUT A General Factoring Strategy It is important to be able to recognize the
More informationChapter 5 Self-Assessment
Chapter 5 Self-Assessment. BLM 5 1 Concept BEFORE DURING (What I can do) AFTER (Proof that I can do this) 5.1 I can multiply binomials. I can multiply trinomials. I can explain how multiplication of binomials
More informationChapter 8: Factoring Polynomials. Algebra 1 Mr. Barr
p. 1 Chapter 8: Factoring Polynomials Algebra 1 Mr. Barr Name: p. 2 Date Schedule Lesson/Activity 8.1 Monomials & Factoring 8.2 Using the Distributive Property 8.3 Quadratics in the form x 2 +bx+c Quiz
More informationFactoring completely is factoring a product down to a product of prime factors. 24 (2)(12) (2)(2)(6) (2)(2)(2)(3)
Factoring Contents Introduction... 2 Factoring Polynomials... 4 Greatest Common Factor... 4 Factoring by Grouping... 5 Factoring a Trinomial with a Table... 5 Factoring a Trinomial with a Leading Coefficient
More informationAccuplacer Review Workshop. Intermediate Algebra. Week Four. Includes internet links to instructional videos for additional resources:
Accuplacer Review Workshop Intermediate Algebra Week Four Includes internet links to instructional videos for additional resources: http://www.mathispower4u.com (Arithmetic Video Library) http://www.purplemath.com
More informationSection 5.3 Factor By Grouping
Section 5.3 Factor By Grouping INTRODUCTION In the previous section you were introduced to factoring out a common monomial factor from a polynomial. For example, in the binomial 6x 2 + 15x, we can recognize
More informationMath 101, Basic Algebra Author: Debra Griffin
Math 101, Basic Algebra Author: Debra Griffin Name Chapter 5 Factoring 5.1 Greatest Common Factor 2 GCF, factoring GCF, factoring common binomial factor 5.2 Factor by Grouping 5 5.3 Factoring Trinomials
More informationSection R.4 Review of Factoring. Factoring Out the Greatest Common Factor
1 Section R.4 Review of Factoring Objective #1: Factoring Out the Greatest Common Factor The Greatest Common Factor (GCF) is the largest factor that can divide into the terms of an expression evenly with
More informationMultiply the binomials. Add the middle terms. 2x 2 7x 6. Rewrite the middle term as 2x 2 a sum or difference of terms. 12x 321x 22
Section 5.5 Factoring Trinomials 349 Factoring Trinomials 1. Factoring Trinomials: AC-Method In Section 5.4, we learned how to factor out the greatest common factor from a polynomial and how to factor
More informationIn this section we revisit two special product forms that we learned in Chapter 5, the first of which was squaring a binomial.
5B. SPECIAL PRODUCTS 11 5b Special Products Special Forms In this section we revisit two special product forms that we learned in Chapter 5, the first of which was squaring a binomial. Squaring a binomial.
More informationName: Algebra Unit 7 Polynomials
Name: Algebra Unit 7 Polynomials Monomial Binomial Trinomial Polynomial Degree Term Standard Form 1 ((2p 3 + 6p 2 + 10p) + (9p 3 + 11p 2 + 3p) TO REMEMBER Adding and Subtracting Polynomials TO REMEMBER
More information5.6 Special Products of Polynomials
5.6 Special Products of Polynomials Learning Objectives Find the square of a binomial Find the product of binomials using sum and difference formula Solve problems using special products of polynomials
More informationSection 5.6 Factoring Strategies
Section 5.6 Factoring Strategies INTRODUCTION Let s review what you should know about factoring. (1) Factors imply multiplication Whenever we refer to factors, we are either directly or indirectly referring
More informationUnit 8: Quadratic Expressions (Polynomials)
Name: Period: Algebra 1 Unit 8: Quadratic Expressions (Polynomials) Note Packet Date Topic/Assignment HW Page Due Date 8-A Naming Polynomials and Combining Like Terms 8-B Adding and Subtracting Polynomials
More informationSolution: To simplify this we must multiply the binomial by itself using the FOIL method.
Special Products This section of notes will focus on the use of formulas to find products. Although it may seem like a lot of extra memorizing, these formulas will save considerable time when multiplying
More informationSection R.5 Review of Factoring. Factoring Out the Greatest Common Factor
1 Section R.5 Review of Factoring Objective #1: Factoring Out the Greatest Common Factor The Greatest Common Factor (GCF) is the largest factor that can divide into the terms of an expression evenly with
More informationElementary Algebra Review for Exam 3
Elementary Algebra Review for Exam ) After receiving a discount of 5% on its bulk order of typewriter ribbons, John's Office Supply pays $5882. What was the price of the order before the discount? Round
More informationSection 1.5: Factoring Special Products
Objective: Identify and factor special products including a difference of two perfect squares, perfect square trinomials, and sum and difference of two perfect cubes. When factoring there are a few special
More informationFactors of 10 = = 2 5 Possible pairs of factors:
Factoring Trinomials Worksheet #1 1. b 2 + 8b + 7 Signs inside the two binomials are identical and positive. Factors of b 2 = b b Factors of 7 = 1 7 b 2 + 8b + 7 = (b + 1)(b + 7) 2. n 2 11n + 10 Signs
More informationLesson 7.1: Factoring a GCF
Name Lesson 7.1: Factoring a GCF Date Algebra I Factoring expressions is one of the gateway skills that is necessary for much of what we do in algebra for the rest of the course. The word factor has two
More information(8m 2 5m + 2) - (-10m 2 +7m 6) (8m 2 5m + 2) + (+10m 2-7m + 6)
Adding Polynomials Adding & Subtracting Polynomials (Combining Like Terms) Subtracting Polynomials (if your nd polynomial is inside a set of parentheses). (x 8x + ) + (-x -x 7) FIRST, Identify the like
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction
Prerequisite Skills This lesson requires the use of the following skills: multiplying polynomials working with complex numbers Introduction 2 b 2 A trinomial of the form x + bx + that can be written as
More informationPrerequisites. Introduction CHAPTER OUTLINE
Prerequisites 1 Figure 1 Credit: Andreas Kambanls CHAPTER OUTLINE 1.1 Real Numbers: Algebra Essentials 1.2 Exponents and Scientific Notation 1.3 Radicals and Rational Expressions 1.4 Polynomials 1.5 Factoring
More informationxyz Degree is 5. See last term.
THE PERFECT SQUARE - COLLEGE ALGEBRA LECTURES Coprights and Author: Kevin Pinegar Chapter 0 PRE-ALGEBRA TOPICS 0.4 Polnomials and Factoring Polnomials And Monomials A monomial is a number, variable or
More informationName Class Date. Adding and Subtracting Polynomials
8-1 Reteaching Adding and Subtracting Polynomials You can add and subtract polynomials by lining up like terms and then adding or subtracting each part separately. What is the simplified form of (3x 4x
More informationAlgebra Module A33. Factoring - 2. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Algebra Module A33 Factoring - 2 Copyright This publication The Northern Alberta Institute of Technology 2002. All Rights Reserved. LAST REVISED November, 2008 Factoring - 2 Statement of Prerequisite
More informationChapter 2 Algebra Part 1
Chapter 2 Algebra Part 1 Section 2.1 Expansion (Revision) In Mathematics EXPANSION really means MULTIPLY. For example 3(2x + 4) can be expanded by multiplying them out. Remember: There is an invisible
More informationSimplifying and Combining Like Terms Exponent
Simplifying and Combining Like Terms Exponent Coefficient 4x 2 Variable (or Base) * Write the coefficients, variables, and exponents of: a) 8c 2 b) 9x c) y 8 d) 12a 2 b 3 Like Terms: Terms that have identical
More informationPolynomials. Factors and Greatest Common Factors. Slide 1 / 128. Slide 2 / 128. Slide 3 / 128. Table of Contents
Slide 1 / 128 Polynomials Table of ontents Slide 2 / 128 Factors and GF Factoring out GF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring
More informationFactoring is the process of changing a polynomial expression that is essentially a sum into an expression that is essentially a product.
Ch. 8 Polynomial Factoring Sec. 1 Factoring is the process of changing a polynomial expression that is essentially a sum into an expression that is essentially a product. Factoring polynomials is not much
More informationMath 154 :: Elementary Algebra
Math 1 :: Elementar Algebra Section.1 Exponents Section. Negative Exponents Section. Polnomials Section. Addition and Subtraction of Polnomials Section. Multiplication of Polnomials Section. Division of
More information2.01 Products of Polynomials
2.01 Products of Polynomials Recall from previous lessons that when algebraic expressions are added (or subtracted) they are called terms, while expressions that are multiplied are called factors. An algebraic
More information7.1 Simplifying Rational Expressions
7.1 Simplifying Rational Expressions LEARNING OBJECTIVES 1. Determine the restrictions to the domain of a rational expression. 2. Simplify rational expressions. 3. Simplify expressions with opposite binomial
More informationPolynomial is a general description on any algebraic expression with 1 term or more. To add or subtract polynomials, we combine like terms.
Polynomials Lesson 5.0 Re-Introduction to Polynomials Let s start with some definition. Monomial - an algebraic expression with ONE term. ---------------------------------------------------------------------------------------------
More informationSpecial Binomial Products
Lesson 11-6 Lesson 11-6 Special Binomial Products Vocabulary perfect square trinomials difference of squares BIG IDEA The square of a binomial a + b is the expression (a + b) 2 and can be found by multiplying
More informationUniversity of Phoenix Material
1 University of Phoenix Material Factoring and Radical Expressions The goal of this week is to introduce the algebraic concept of factoring polynomials and simplifying radical expressions. Think of factoring
More informationIs the following a perfect cube? (use prime factorization to show if it is or isn't) 3456
Is the following a perfect cube? (use prime factorization to show if it is or isn't) 3456 Oct 2 1:50 PM 1 Have you used algebra tiles before? X 2 X 2 X X X Oct 3 10:47 AM 2 Factor x 2 + 3x + 2 X 2 X X
More information5.1 Exponents and Scientific Notation
5.1 Exponents and Scientific Notation Definition of an exponent a r = Example: Expand and simplify a) 3 4 b) ( 1 / 4 ) 2 c) (0.05) 3 d) (-3) 2 Difference between (-a) r (-a) r = and a r a r = Note: The
More informationFactoring is the process of changing a polynomial expression that is essentially a sum into an expression that is essentially a product.
Ch. 8 Polynomial Factoring Sec. 1 Factoring is the process of changing a polynomial expression that is essentially a sum into an expression that is essentially a product. Factoring polynomials is not much
More informationTool 1. Greatest Common Factor (GCF)
Chapter 7: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationMini-Lecture 6.1 The Greatest Common Factor and Factoring by Grouping
Copyright 01 Pearson Education, Inc. Mini-Lecture 6.1 The Greatest Common Factor and Factoring by Grouping 1. Find the greatest common factor of a list of integers.. Find the greatest common factor of
More informationStudy P.5 CVC 1 7, # 1, 5, 9,...37, 39 55, 59, 65, 69, 73,
GOALS: Factor Polynomials using: 1. Distributive Property (common factors) 2. Trial and Error (trinomials) 3. Factor by Grouping (trinomials) Study P.5 CVC 1 7, # 1, 5, 9,...37, 39 55, 59, 65, 69, 73,...
More informationPolynomials. Unit 10 Polynomials 2 of 2 SMART Board Notes.notebook. May 15, 2013
Oct 19 9:41 M errick played basketball for 5 out of the 10 days for four hours each. How many hours did errick spend playing basketball? Oct 19 9:41 M Polynomials Polynomials 1 Table of ontents Factors
More informationChapter 5 Polynomials
Department of Mathematics Grossmont College October 7, 2012 Multiplying Polynomials Multiplying Binomials using the Distributive Property We can multiply two binomials using the Distributive Property,
More informationSection 7.1 Common Factors in Polynomials
Chapter 7 Factoring How Does GPS Work? 7.1 Common Factors in Polynomials 7.2 Difference of Two Squares 7.3 Perfect Trinomial Squares 7.4 Factoring Trinomials: (x 2 + bx + c) 7.5 Factoring Trinomials: (ax
More informationMTH 110-College Algebra
MTH 110-College Algebra Chapter R-Basic Concepts of Algebra R.1 I. Real Number System Please indicate if each of these numbers is a W (Whole number), R (Real number), Z (Integer), I (Irrational number),
More informationThe two meanings of Factor 1. Factor (verb) : To rewrite an algebraic expression as an equivalent product
At the end of Packet #1we worked on multiplying monomials, binomials, and trinomials. What we have to learn now is how to go backwards and do what is called factoring. The two meanings of Factor 1. Factor
More information1. Which pair of factors of 8 has a sum of 9? 1 and 8 2. Which pair of factors of 30 has a sum of. r 2 4r 45
Warm Up 1. Which pair of factors of 8 has a sum of 9? 1 and 8 2. Which pair of factors of 30 has a sum of 17? 2 and 15 Multiply. 3. (x +2)(x +3) x 2 + 5x + 6 4. (r + 5)(r 9) r 2 4r 45 Objective Factor
More information2-4 Completing the Square
2-4 Completing the Square Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Write each expression as a trinomial. 1. (x 5) 2 x 2 10x + 25 2. (3x + 5) 2 9x 2 + 30x + 25 Factor each expression. 3.
More informationIdentifying & Factoring: x 2 + bx + c
Identifying & Factoring: x 2 + bx + c Apr 13 11:04 AM 1 May 16 8:52 AM 2 A polynomial that can be simplified to the form ax + bx + c, where a 0, is called a quadratic polynomial. Linear term. Quadratic
More informationAdding and Subtracting Rational Expressions
Adding and Subtracting Rational Expressions To add or subtract rational expressions, follow procedures similar to those used in adding and subtracting rational numbers. 4 () 4(3) 10 1 3 3() (3) 1 1 1 All
More informationUnit: Polynomials and Factoring
Unit: Polynomials: Multiplying and Factoring Name Dates Taught Specific Outcome 10I.A.1 Demonstrate an understanding of factors of whole numbers by determining: Prime factors Greatest common factor Least
More informationFactor Trinomials When the Coefficient of the Second-Degree Term is 1 (Objective #1)
Factoring Trinomials (5.2) Factor Trinomials When the Coefficient of the Second-Degree Term is 1 EXAMPLE #1: Factor the trinomials. = = Factor Trinomials When the Coefficient of the Second-Degree Term
More informationSection 7.4 Additional Factoring Techniques
Section 7.4 Additional Factoring Techniques Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Factor trinomials when a = 1. Multiplying binomials
More informationMathematics 10C. UNIT THREE Polynomials. 3x 3-6x 2. 3x 2 (x - 2) 4x 2-3x - 1. Unit. Student Workbook. FOIL (2x - 3)(x + 1) A C = -4.
Mathematics 10C FOIL (2x - 3)(x + 1) Student Workbook Lesson 1: Expanding Approximate Completion Time: 4 Days Unit 3 3x 3-6x 2 Factor Expand 3x 2 (x - 2) Lesson 2: Greatest Common Factor Approximate Completion
More informationWeek 20 Algebra 1 Assignment:
Week 0 Algebra 1 Assignment: Day 1: pp. 38-383 #-0 even, 3-7 Day : pp. 385-386 #-18 even, 1-5 Day 3: pp. 388-389 #-4 even, 7-9 Day 4: pp. 39-393 #1-37 odd Day 5: Chapter 9 test Notes on Assignment: Pages
More informationS3 (3.1) Mutiplying out brackets & Factorising.notebook February 09, 2016
Daily Practice 30.11.15 Q1. State the equation of the line that passes through (0, 8) and (3, 1) Q2. Simplify 500 Today we will be marking the check-up, homework and revising over multiplying out and simplifying.
More informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson 10.1 Name Date Water Balloons Polynomials and Polynomial Functions Vocabulary Match each key term to its corresponding definition. 1. A polynomial written with
More informationFactoring Quadratic Expressions VOCABULARY
5-5 Factoring Quadratic Expressions TEKS FOCUS Foundational to TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil,
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationChapter 6: Quadratic Functions & Their Algebra
Chapter 6: Quadratic Functions & Their Algebra Topics: 1. Quadratic Function Review. Factoring: With Greatest Common Factor & Difference of Two Squares 3. Factoring: Trinomials 4. Complete Factoring 5.
More informationChapter 6.1: Introduction to parabolas and solving equations by factoring
Chapter 6 Solving Quadratic Equations and Factoring Chapter 6.1: Introduction to parabolas and solving equations by factoring If you push a pen off a table, how does it fall? Does it fall like this? Or
More informationFactoring Trinomials of the Form
Section 7 3: Factoring Trinomials of the Form 1x 2 + Bx + C The FOIL process changes a product of 2 binomials into a polynomial. The reverse process starts with a polynomial and finds the 2 binomials whose
More informationSection 13.1 The Greatest Common Factor and Factoring by Grouping. to continue. Also, circle your answer to each numbered exercise.
Algebra Foundations First Edition, Elayn Martin-Gay Sec. 13.1 Section 13.1 The Greatest Common Factor and Factoring by Grouping Complete the outline as you view Video Lecture 13.1. Pause the video as needed
More informationFactoring Methods. Example 1: 2x * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More informationWhen Is Factoring Used?
When Is Factoring Used? Name: DAY 9 Date: 1. Given the function, y = x 2 complete the table and graph. x y 2 1 0 1 2 3 1. A ball is thrown vertically upward from the ground according to the graph below.
More informationChapter 4 Factoring and Quadratic Equations
Chapter 4 Factoring and Quadratic Equations Lesson 1: Factoring by GCF, DOTS, and Case I Lesson : Factoring by Grouping & Case II Lesson 3: Factoring by Sum and Difference of Perfect Cubes Lesson 4: Solving
More information3.1 Factors and Multiples of Whole Numbers
3.1 Factors and Multiples of Whole Numbers LESSON FOCUS: Determine prime factors, greatest common factors, and least common multiples of whole numbers. The prime factorization of a natural number is the
More informationChapter 4 Partial Fractions
Chapter 4 8 Partial Fraction Chapter 4 Partial Fractions 4. Introduction: A fraction is a symbol indicating the division of integers. For example,, are fractions and are called Common 9 Fraction. The dividend
More informationSection 13-1: The Distributive Property and Common Factors
Section 13-1: The Distributive Property and Common Factors Factor: 4y 18z 4y 18z 6(4y 3z) Identify the largest factor that is common to both terms. 6 Write the epression as a product by dividing each term
More information(2/3) 3 ((1 7/8) 2 + 1/2) = (2/3) 3 ((8/8 7/8) 2 + 1/2) (Work from inner parentheses outward) = (2/3) 3 ((1/8) 2 + 1/2) = (8/27) (1/64 + 1/2)
Exponents Problem: Show that 5. Solution: Remember, using our rules of exponents, 5 5, 5. Problems to Do: 1. Simplify each to a single fraction or number: (a) ( 1 ) 5 ( ) 5. And, since (b) + 9 + 1 5 /
More information