# Summer Prep Packet for students entering Algebra 2

Size: px
Start display at page:

Download "Summer Prep Packet for students entering Algebra 2"

Transcription

1 Summer Prep Packet for students entering Algebra The following skills and concepts included in this packet are vital for your success in Algebra. The Mt. Hebron Math Department encourages all students to keep up with their math skills over the summer. We believe that worksheets contained in this summer packet will be helpful to you as you prepare for a great math year in Algebra. *Please note: Completion of the following packet is strongly encouraged although it is not a requirement.

2 Solving Multi-Step Equations Solve Multi-Step Equations To solve equations with more than one operation, often called multi-step equations, undo operations by working backward. Reverse the usual order of operations as you work. Example Solve 5x + 3 = 3. 5x + 3 = 3 5x = 3-3 5x = 0 5x 5 = 0 5 x = 4 Original equation. Subtract 3 from each side. Simplify. Divide each side by 5. Simplify. Solve each equation. Check your solution. 1. 5x + = 7. 6x + 9 = x + 16 = n - 8 = x = p - 4 = = d n 1 = g = b + 8 = x - 8 = y = = 7x - (-1) = -1 + k -4 Write an equation and solve each problem. 16. Find three consecutive integers whose sum is Find two consecutive odd integers whose sum is Find three consecutive integers whose sum is = 10y - 40 Chapter Glencoe Algebra 1

3 Solving Equations with the Variable on Each Side Variables on Each Side To solve an equation with the same variable on each side, first use the Addition or the Subtraction Property of Equality to write an equivalent equation that has the variable on just one side of the equation. Then solve the equation. Example 1 Solve 5y - 8 = 3y + 1. Example 5y - 8 = 3y + 1 5y - 8-3y = 3y + 1-3y y - 8 = 1 y = y = 0 y = 0 y = 10 The solution is 10. Solve -11-3y = 8y y = 8y y + 3y = 8y y -11 = 11y = 11y = 11y = 11y = y The solution is Solve each equation. Check your solution b = 5b y - y = 3y x + = x n - 8 = 3n x =.1 - x m + 6. = 8.8m b + 4 = 1 8 b k - 5 = 1 k p = 4p b - 8 = 10 - b x - 8 = - - x 1. 3y = 3y x = 7x k = k a = 10a n + 8 = 1 n y - 8 = 9-3 y r + 5 = 5-4r x = 6x k = -10-4k y = 10y - 1 Chapter 3 Glencoe Algebra 1

4 Multiplying Monomials Monomials A monomial is a number, a variable, or the product of a number and one or more variables with nonnegative integer exponents. An expression of the form x n is called a power and represents the product you obtain when x is used as a factor n times. To multiply two powers that have the same base, add the exponents. Product of Powers For any number a and all integers m and n, a m a n = a m + n. Example 1 Simplify (3x 6 )(5x ). Example (3x 6 )(5x ) = (3)(5)(x 6 x ) Group the coeffi cients and the variables = (3 5)(x 6 + ) Product of Powers = 15x 8 Simplify. The product is 15x 8. Simplify each expression. Simplify (-4a 3 b)(3a b 5 ). (-4a 3 b)(3a b 5 ) = (-4)(3)(a 3 a )(b b 5 ) = -1(a 3 + )(b ) = -1a 5 b 6 The product is -1a 5 b y(y 5 ). n n 7 3. (-7x )(x 4 ) 4. x(x )(x 4 ) 5. m m 5 6. (-x 3 )(-x 4 ) 7. (a )(8a) 8. (rn)(rn 3 )(n ) 9. (x y)(4xy 3 ) ( a 3 b)( 6b 3 ) 11. (-4x 3 )(-5x 7 ) 1. (-3j k 4 )(jk 6 ) 13. ( 5a bc 3 ) ( 1 5 abc 4 ) 14. (-5xy)(4x )(y 4 ) 15. (10x 3 yz )(-xy 5 z) Chapter 7 89 Glencoe Algebra 1

5 Multiplying Monomials Simplify Expressions An expression of the form (x m ) n is called a power of a power and represents the product you obtain when x m is used as a factor n times. To find the power of a power, multiply exponents. Power of a Power For any number a and all integers m and n, (a m ) n = a mn. Power of a Product For any number a and all integers m and n, (ab) m = a m b m. We can combine and use these properties to simplify expressions involving monomials. Example Simplify (-ab ) 3 (a ) 4. (-ab ) 3 (a ) 4 = (-ab ) 3 (a 8 ) Power of a Power = (-) 3 (a 3 )(b ) 3 (a 8 ) Power of a Product = (-) 3 (a 3 )(a 8 )(b ) 3 Group the coefficients and the variables = (-) 3 (a 11 )(b ) 3 Product of Powers = -8a 11 b 6 Power of a Power The product is -8a 11 b 6. Simplify each expression. 1. (y 5 ). (n 7 ) 4 3. (x ) 5 (x 3 ) 4. -3(ab 4 ) 3 5. (-3ab 4 ) 3 6. (4x b) 3 7. (4a ) (b 3 ) 8. (4x) (b 3 ) 9. (x y 4 ) (a 3 b )(b 3 ) 11. (-4xy) 3 (-x ) 3 1. (-3j k 3 ) (j k) (5 a b) 3 ( 1 5 abf ) 14. (xy) (-3x )(4y 4 ) 15. (x 3 y z ) 3 (x z) (-n 6 y 5 )(-6n 3 y )(ny) (-3a 3 n 4 )(-3a 3 n) (x) 4 (4x 5 y) Chapter 7 90 Glencoe Algebra 1

6 Dividing Monomials Quotients of Monomials To divide two powers with the same base, subtract the exponents. Quotient of Powers Power of a Quotient For all integers m and n and any nonzero number a, a m a n = a m-n. For any integer m and any real numbers a and b, b 0, ( a b) m = a m b m. Example 1 Simplify a 4 b 7 ab. Assume Example Simplify ( a 3 b 5 3 b ) 3. Assume that no denominator equals zero. a 4 b 7 ab = ( a 4 ) that no denominator equals zero. a ( b 7 b ) Group powers with the same base. ( a 3 b 5 3 b ) 3 = ( a 3 b 5 ) 3 Power of a Quotient (3 b 3 ) = (a 4-1 )(b 7 - ) Quotient of Powers = a 3 b 5 = 3 ( a 3 ) 3 ( b 5 ) 3 Power of a Product Simplify. (3) 3 ( b ) 3 The quotient is a 3 b 5. = 8 a 9 b 15 Power of a Power 7 b 6 = 8 a 9 b 9 7 The quotient is 8 a 9 b 9 7. Quotient of Powers Simplify each expression. Assume that no denominator equals zero m 6 m 4 4. a a 5. x 5 y 3 x 5 y 7. x y 6 y 4 x 10. ( r 5 w 3 8. ( a b r 4 w ) ( 3 r 6 n p 5 n 4 p n 6. - y 7 14 y 5 ) 3 a 9. ( 4 p 4 r 4 3 p r ) r 5 n ) 4 1. r 7 n 7 t n 3 r 3 t 3 Chapter 7 91 Glencoe Algebra 1

7 Multiplying Polynomials Multiply Binomials To multiply two binomials, you can apply the Distributive Property twice. A useful way to keep track of terms in the product is to use the FOIL method as illustrated in Example. Example 1 Find (x + 3)(x - 4). Example Find (x - )(x + 5) using Horizontal Method (x + 3)(x - 4) the FOIL method. (x - )(x + 5) = x(x - 4) + 3(x - 4) = (x)(x) + x(-4) + 3(x)+ 3(-4) = x - 4x + 3x - 1 = x - x - 1 Vertical Method x + 3 ( ) x - 4-4x - 1 x + 3x x - x - 1 The product is x - x - 1. First Outer Inner Last = (x)(x) + (x)(5) + (-)(x) + (-)(5) = x + 5x + (-x) - 10 = x + 3x - 10 The product is x + 3x Find each product. 1. (x + )(x + 3). (x - 4)(x + 1) 3. (x - 6)(x - ) 4. ( p - 4)( p + ) 5. ( y + 5)( y + ) 6. (x - 1)(x + 5) 7. (3n - 4)(3n - 4) 8. (8m - )(8m + ) 9. (k + 4)(5k - 1) 10. (3x + 1)(4x + 3) 11. (x - 8)(-3x + 1) 1. (5t + 4)(t - 6) 13. (5m - 3n)(4m - n) 14. (a - 3b)(a - 5b) 15. (8x - 5)(8x + 5) 16. (n - 4)(n + 5) 17. (4m - 3)(5m - 5) 18. (7g - 4)(7g + 4) Chapter Glencoe Algebra 1

8 Using the Distributive Property Solve Equations by Factoring The following property, along with factoring, can be used to solve certain equations. Zero Product Property For any real numbers a and b, if ab = 0, then either a = 0, b = 0, or both a and b equal 0. Example Solve 9x + x = 0. Then check the solutions. Write the equation so that it is of the form ab = 0. 9x + x = 0 Original equation x(9x + 1) = 0 Factor the GCF of 9x + x, which is x. x = 0 or 9x + 1 = 0 x = 0 x = The solution set is {0, - 1 9}. Zero Product Property Solve each equation. Check Substitute 0 and - 1 for x in the original equation. 9 9x + x = 0 9x + x = 0 9(0) (- 1 9) + ( ) 1 0 = ( ) 0 = 0 Solve each equation. Check your solutions. 1. x(x + 3) = 0. 3m(m - 4) = 0 3. (r - 3)(r + ) = x(x - 1) = 0 5. (4m + 8)(m - 3) = t = 5t 7. (4c + )(c - 7) = p - 15p = y = 8y 10. 1x = -6x 11. (4a + 3)(8a + 7) = y = 1y 13. x = -x 14. (6y - 4)(y + 3) = m = 4m 16. 1x = 3x 17. 1a = -3a 18. (1a + 4)(3a - 1) = 0 Chapter Glencoe Algebra 1

9 Quadratic Equations: x + bx + c = 0 Factor x + bx + c To factor a trinomial of the form x + bx + c, find two integers, m and p, whose sum is equal to b and whose product is equal to c. Factoring x + bx + c x + bx + c = (x + m)(x + p), where m + p = b and mp = c. Example 1 Factor each polynomial. Example Factor x + 6x a. x + 7x + 10 In this trinomial, b = 7 and c = 10. Factors of 10 Sum of Factors 1, 10 11, 5 7 Since + 5 = 7 and 5 = 10, let m = and p = 5. x + 7x + 10 = (x + 5)(x + ) b. x - 8x + 7 In this trinomial, b = -8 and c = 7. Notice that m + p is negative and mp is positive, so m and p are both negative. Since -7 + (-1) = -8 and (-7)(-1) = 7, m = -7 and p = -1. x - 8x + 7 = (x - 7)(x - 1) Factor each polynomial. In this trinomial, b = 6 and c = -16. This means m + p is positive and mp is negative. Make a list of the factors of -16, where one factor of each pair is positive. Factors of -16 Sum of Factors 1, , 16 15, , 8 6 Therefore, m = - and p = 8. x + 6x - 16 = (x - )(x + 8) 1. x + 4x + 3. m + 1m r - 3r + 4. x - x x - 4x x - x t - 4t p - 16p x + x 10. x + 6x a + 8a y - 7y x - x y + 14y m + 9m x + 1x a - 14a y + y 19. x + xy + y 0. a - 4ab + 4b 1. x + 6xy - 7y Chapter Glencoe Algebra 1

10 Name: Factoring Quadratics with a=1 Date: Pd: Directions: Factor each polynomial completely (you may need to factor the GCF first). All work should be neat and organized on a separate sheet of paper. Circle / box your final answer. 1. u - 1u. 6x + x 3. z + 10z x + 14x q 9q k 4k 3 7. b + 18b m + 6m 7 9. p 5p n 7n j 1j h + 8h b + b a 3a y 3y r + 14r a x 8x x + 7x t + 15t 1. w 8w + 1. c 5c r 35r 4. z z 4 5. t + 1t a a 6a p w 5/4 30. t 6t x x + 5x c + 6c b 9b y -11y x x p + 19p h g 1g t 41. j 10j x x + 11 BONUS: 43. 3z + 4z y 5y - 1

11 Quadratic Equations: x + bx + c = 0 Solve Equations by Factoring Factoring and the Zero Product Property can be used to solve many equations of the form x + bx + c = 0. Example 1 Solve x + 6x = 7. Check your solutions. x + 6x = 7 Original equation x + 6x - 7 = 0 Rewrite equation so that one side equals 0. (x - 1)(x + 7) = 0 Factor. x - 1 = 0 or x + 7 = 0 Zero Product Property x = 1 x = -7 Solve each equation. The solution set is {1, -7}. Since = 7 and (-7) + 6(-7) = 7, the solutions check. Example ROCKET LAUNCH A rocket is fired with an initial velocity of 88 feet per second. How many seconds will it take for the rocket to hit the ground? The formula h = vt - 16t gives the height h of the rocket after t seconds when the initial velocity v is given in feet per second. h = vt - 16t Formula 0 = 88t - 16t Substitute. 0 = 16t(143 - t) Factor. 16t = 0 or t = 0 Zero Product Property t = 0 t = 143 Solve each equation. The value t = 0 represents the time at launch. The rocket returns to the ground in 143 seconds, or a little less than.5 minutes after launch. Solve each equation. Check the solutions. 1. x - 4x + 3 = 0. y - 5y + 4 = 0 3. m + 10m + 9 = 0 4. x = x + 5. x - 4x = 5 6. x - 1x + 36 = 0 7. t - 8 = -7t 8. p = 9p x + x = x + 6 = 5x 11. a = 11a y - 8y + 15 = x = 4-10x 14. a - 18a = b = 10b - 16 Use the formula h = vt - 16t to solve each problem. 16. FOOTBALL A punter can kick a football with an initial velocity of 48 feet per second. How many seconds will it take for the ball to return to the ground? 17. BASEBALL A ball is thrown up with an initial velocity of 3 feet per second. How many seconds will it take for the ball to return to the ground? 18. ROCKET LAUNCH If a rocket is launched with an initial velocity of 1600 feet per second, when will the rocket be 14,400 feet high? Chapter Glencoe Algebra 1

12 Solving Quadratic Equations by Using the Quadratic Formula Quadratic Formula To solve the standard form of the quadratic equation, ax + bx + c = 0, use the Quadratic Formula. Quadratic Formula -b ± the formula x = b - 4ac a that gives the solutions of ax + bx + c = 0, where a 0 Example 1 Solve x + x = 3 by Example Solve x - 6x - = 0 by using the Quadratic Formula. Rewrite the equation in standard form. using the Quadratic Formula. Round to the nearest tenth if necessary. x + x = 3 Original equation For this equation a = 1, b = -6, and c = -. x + x - 3 = 3-3 Subtract 3 from each side. x = -b ± b - 4ac x + x - 3 = 0 Simplify. a Now let a = 1, b =, and c = -3 in the Quadratic Formula. x = -b ± b - 4ac a a = - ± () - 4(1)(-3) (1) - + = 16 x = or x = = 1 = -3 The solution set is {-3, 1}. = 6 ± (-6) - 4(1)(-) (1) = 6 ± 44 x = 6 ± 44 or x = 6-44 x The solution set is {-0.3, 6.3}. Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 1. x - 3x + = 0. x - 8x = x - 8x = x + 5x = x + x = x - 8x - 5 = x + 19x = 1 8. x + 6x = x + x - 15 = x - 4x = x + 5x = x + 9x - 4 = x + 9x + 4 = x + 17x + = 0 Chapter 9 15 Glencoe Algebra 1

13

14 Simplifying Radical Expressions Product Property of Square Roots The Product Property of Square Roots and prime factorization can be used to simplify expressions involving irrational square roots. When you simplify radical expressions with variables, use absolute value to ensure nonnegative results. Product Property of Square Roots For any numbers a and b, where a 0 and b 0, ab = a b. Example 1 Simplify = Prime factorization of 180 = 3 5 Product Property of Square Roots = 3 5 Simplify. = 6 5 Simplify. Example 10a b 5 c 4 Simplify 10a b 5 c 4. = a b 5 c 4 = 3 5 a b 4 b c 4 = 3 5 a b b c = a b c 30b Simplify each expression a 10. 9x x 3 3x a 4 b x 4 y a a b a b c 1. 18c x 3 y 0. 7a 6 b 3 c 1. 45x y 5 z 8. 98x 4 y 6 z Chapter Glencoe Algebra 1

### Algebra I. Polynomials.

1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying

### Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals

Algebra 1 Math Review Packet Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals 2017 Math in the Middle 1. Clear parentheses using the distributive

### Algebra I Polynomials

Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying

### Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is

### Solving Multi-Step Equations

1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract terms to both sides of the equation to get the

### LESSON 9.1 ROOTS AND RADICALS

LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical

### Basic Fraction and Integer Operations (No calculators please!)

P1 Summer Math Review Packet For Students entering Geometry The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in Geometry.

### review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17

1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

### When factoring, we ALWAYS start with the (unless it s 1).

Math 100 Elementary Algebra Sec 5.1: The Greatest Common Factor and Factor By Grouping (FBG) Recall: In the product XY, X and Y are factors. Defn In an expression, any factor that is common to each term

### Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +

### Herndon High School Geometry Honors Summer Assignment

Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in

### Algebra I Unit Report Summary

Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02

### 5.3. Polynomials and Polynomial Functions

5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a

### LESSON 7.2 FACTORING POLYNOMIALS II

LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II 305 OVERVIEW Here s what you ll learn in this lesson: Trinomials I a. Factoring trinomials of the form x 2 + bx + c; x 2 + bxy +

### Gaithersburg High School Summer 2018 Math Packet For Rising Algebra 2/Honors Algebra 2 Students

Gaithersburg High School Math Packet For Rising Algebra 2/Honors Algebra 2 Students 1 This packet is an optional review of the skills that will help you be successful in Algebra 2 in the fall. By completing

### Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

### Variables and Expressions

Variables and Expressions A variable is a letter that represents a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic

### Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

### 27 Wyner Math 2 Spring 2019

27 Wyner Math 2 Spring 2019 CHAPTER SIX: POLYNOMIALS Review January 25 Test February 8 Thorough understanding and fluency of the concepts and methods in this chapter is a cornerstone to success in the

### Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3

Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring

### Multiplication of Polynomials

Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is

### Answer Key. Solve each equation x - 9 = (n + 2) = b - 6 = -3b + 48

Solve each equation. 1. -3x - 9 = -27 2. 25 + 2(n + 2) = 30 3. -9b - 6 = -3b + 48 x = 6 n = 1 / 2 b = -9 4. 5 - (m - 4) = 2m + 3(m - 1) 5. -24-10k = -8(k + 4) - 2k 6. f - (-19) = 11f + 23-20f m = 2 no

### Study Guide for Math 095

Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.

### Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in

### MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017

Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

### Geometry Summer Assignment 2018

Geometry Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Geometry this year. You are advised to be familiar with each

### A-2. Polynomials and Factoring. Section A-2 1

A- Polynomials and Factoring Section A- 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring

### HONORS GEOMETRY Summer Skills Set

HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference

### Math 0320 Final Exam Review

Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:

### CP Algebra 2 Unit 2-1: Factoring and Solving Quadratics WORKSHEET PACKET

CP Algebra Unit -1: Factoring and Solving Quadratics WORKSHEET PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor

### Beginning Algebra MAT0024C. Professor Sikora. Professor M. J. Sikora ~ Valencia Community College

Beginning Algebra Professor Sikora MAT002C POLYNOMIALS 6.1 Positive Integer Exponents x n = x x x x x [n of these x factors] base exponent Numerical: Ex: - = where as Ex: (-) = Ex: - = and Ex: (-) = Rule:

### Algebra 1B Final Review

Name: Class: Date: ID: A Algebra 1B Final Review Short Answer 1. Originally a rectangle was twice as long as it was wide. When 5 m was subtracted from its length and 3 m subtracted from its width, the

### Algebra I. Book 2. Powered by...

Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........

### Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents

Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations

### Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.

Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

### Slide 1 / 200. Quadratic Functions

Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

### When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.

Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules

### Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together

### UNIT 2 FACTORING. M2 Ch 11 all

UNIT 2 FACTORING M2 Ch 11 all 2.1 Polynomials Objective I will be able to put polynomials in standard form and identify their degree and type. I will be able to add and subtract polynomials. Vocabulary

### Sections 7.2, 7.3, 4.1

Sections 7., 7.3, 4.1 Section 7. Multiplying, Dividing and Simplifying Radicals This section will discuss the rules for multiplying, dividing and simplifying radicals. Product Rule for multiplying radicals

### Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions

1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:

### 5.1 Monomials. Algebra 2

. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific

### Rising 8th Grade Math. Algebra 1 Summer Review Packet

Rising 8th Grade Math Algebra 1 Summer Review Packet 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract

### MathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017

Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

### Real Numbers. Real numbers are divided into two types, rational numbers and irrational numbers

Real Numbers Real numbers are divided into two types, rational numbers and irrational numbers I. Rational Numbers: Any number that can be expressed as the quotient of two integers. (fraction). Any number

### P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and

### Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

MCA AP Calculus AB Summer Assignment The following packet is a review of many of the skills needed as we begin the study of Calculus. There two major sections to this review. Pages 2-9 are review examples

### Westside. Algebra 2 PreAP

Westside Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for

### Radicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).

Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the

### New Rochelle High School Geometry Summer Assignment

NAME - New Rochelle High School Geometry Summer Assignment To all Geometry students, This assignment will help you refresh some of the necessary math skills you will need to be successful in Geometry and

### Westside Algebra 2 PreAP

Westside Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for

### MULTIPLYING TRINOMIALS

Name: Date: 1 Math 2 Variable Manipulation Part 4 Polynomials B MULTIPLYING TRINOMIALS Multiplying trinomials is the same process as multiplying binomials except for there are more terms to multiply than

### Algebra Final Exam Review Packet

Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:

### Algebra I CP Final Exam Review

Class: Date: Algebra I CP Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Identify the graph that displays the height of a ping pong

### Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

### SUMMER REVIEW PACKET. Name:

Wylie East HIGH SCHOOL SUMMER REVIEW PACKET For students entering Regular PRECALCULUS Name: Welcome to Pre-Calculus. The following packet needs to be finished and ready to be turned the first week of the

### Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o

Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o o ( 1)(9) 3 ( 1) 3 9 1 Evaluate the second expression at the left, if

### Review Notes - Solving Quadratic Equations

Review Notes - Solving Quadratic Equations What does solve mean? Methods for Solving Quadratic Equations: Solving by using Square Roots Solving by Factoring using the Zero Product Property Solving by Quadratic

### Degree of a polynomial

Variable Algebra Term Polynomial Monomial Binomial Trinomial Degree of a term Degree of a polynomial Linear A generalization of arithmetic. Letters called variables are used to denote numbers, which are

### Properties of Real Numbers

Pre-Algebra Properties of Real Numbers Identity Properties Addition: Multiplication: Commutative Properties Addition: Multiplication: Associative Properties Inverse Properties Distributive Properties Properties

### SUMMER MATH PACKET ALGEBRA TWO COURSE 229

SUMMER MATH PACKET ALGEBRA TWO COURSE 9 MATH SUMMER PACKET INSTRUCTIONS MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The

### Controlling the Population

Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1

### Sharpening your algebra skills: Summer practice for students. Updated 2009

Sharpening your algebra skills: Summer practice for students Updated 009 1 Summer Work: Enhancing Your Algebra Skills in Preparation for a Successful Year at Asheville School The mission of the Asheville

### Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.

Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a

### Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

### Something that can have different values at different times. A variable is usually represented by a letter in algebraic expressions.

Lesson Objectives: Students will be able to define, recognize and use the following terms in the context of polynomials: o Constant o Variable o Monomial o Binomial o Trinomial o Polynomial o Numerical

### Chapter 8 Polynomials and Factoring

Chapter 8 Polynomials and Factoring 8.1 Add and Subtract Polynomials Monomial A. EX: Degree of a monomial the of all of the of the EX: 4x 2 y Polynomial A or EX: Degree of a polynomial the of its terms

### { independent variable some property or restriction about independent variable } where the vertical line is read such that.

Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with

### Unit 13: Polynomials and Exponents

Section 13.1: Polynomials Section 13.2: Operations on Polynomials Section 13.3: Properties of Exponents Section 13.4: Multiplication of Polynomials Section 13.5: Applications from Geometry Section 13.6:

### Florida Math Curriculum (433 topics)

Florida Math 0028 This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular

### Accuplacer Review Workshop. Elementary Algebra Part II. Week Three. Includes internet links to instructional videos for additional resources:

Accuplacer Review Workshop Elementary Algebra Part II Week Three Includes internet links to instructional videos for additional resources: http://www.mathispower4u.com (Arithmetic Video Library) http://www.purplemath.com

### Solving Equations. Solving Equations - decimal coefficients and constants. 2) Solve for x: 3(3x 6) = 3(x -2) 1) Solve for x: 5 x 2 28 x

Level C Review Packet This packet briefly reviews the topics covered on the Level A Math Skills Assessment. If you need additional study resources and/or assistance with any of the topics below, please

### Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring

Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis

### Adding and Subtracting Polynomials

Adding and Subtracting Polynomials Polynomial A monomial or sum of monomials. Binomials and Trinomial are also polynomials. Binomials are sum of two monomials Trinomials are sum of three monomials Degree

### Name: Chapter 7: Exponents and Polynomials

Name: Chapter 7: Exponents and Polynomials 7-1: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You

### Instructor: Richard Getso Course: Math 200.P10 TR 1:00 PM Spring 2016 (Getso)

1/8/016 Practice Test 1 (Chapter 11) Richard Getso Student: Richard Getso Date: 1/8/16 Instructor: Richard Getso Course: Math 00.P10 TR 1:00 PM Spring 016 (Getso) Assignment: Practice Test 1 (Chapter 11)

### = (Type exponential notation with positive exponents)

1. Subtract. Simplify by collecting like radical terms if possible. 2 2 = (Simplify your answer) 2. Add. Simplify if possible. = (Simplify your answer) 3. Divide and simplify. = (Type exponential notation

### Chapter 5: Exponents and Polynomials

Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5

### Math 75 Mini-Mod Due Dates Spring 2016

Mini-Mod 1 Whole Numbers Due: 4/3 1.1 Whole Numbers 1.2 Rounding 1.3 Adding Whole Numbers; Estimation 1.4 Subtracting Whole Numbers 1.5 Basic Problem Solving 1.6 Multiplying Whole Numbers 1.7 Dividing

### The number part of a term with a variable part. Terms that have the same variable parts. Constant terms are also like terms.

Algebra Notes Section 9.1: Add and Subtract Polynomials Objective(s): To be able to add and subtract polynomials. Recall: Coefficient (p. 97): Term of a polynomial (p. 97): Like Terms (p. 97): The number

### Algebra 2/Trig Apps: Chapter 5 Quadratics Packet

Algebra /Trig Apps: Chapter 5 Quadratics Packet In this unit we will: Determine what the parameters a, h, and k do in the vertex form of a quadratic equation Determine the properties (vertex, axis of symmetry,

### Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero

Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,

### correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1

correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 McDougal Littell Algebra 1 2007 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 The main goal of Algebra is to

### Algebra I. Slide 1 / 216. Slide 2 / 216. Slide 3 / 216. Polynomials

Slide 1 / 216 Slide 2 / 216 lgebra I Polynomials 2015-11-02 www.njctl.org Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Multiplying a Polynomial by

### mn 3 17x 2 81y 4 z Algebra I Definitions of Monomials, Polynomials and Degrees 32,457 Slide 1 / 216 Slide 2 / 216 Slide 3 / 216 Slide 4 / 216

Slide 1 / 216 Slide 2 / 216 lgebra I Polynomials 2015-11-02 www.njctl.org Slide 3 / 216 Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Multiplying a

### Solving Quadratic Equations

Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic

### SUMMER ASSIGNMENT FOR ALGEBRA II/TRIGONOMETRY

SUMMER ASSIGNMENT FOR ALGEBRA II/TRIGONOMETRY This summer assignment is designed to ensure that you are prepared for Algebra II/ Trigonometry. Nothing on this summer assignment is new. Everything is a

### CONTENTS COLLEGE ALGEBRA: DR.YOU

1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 1-1 REVIEW A. p. LECTURE 1- RADICALS A.10 p.9 LECTURE 1- COMPLEX NUMBERS A.7 p.17 LECTURE 1-4 BASIC FACTORS A. p.4 LECTURE 1-5. SOLVING THE EQUATIONS A.6 p.

### Algebra III and Trigonometry Summer Assignment

Algebra III and Trigonometry Summer Assignment Welcome to Algebra III and Trigonometry! This summer assignment is a review of the skills you learned in Algebra II. Please bring this assignment with you

### We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:

R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)

### Algebra 31 Summer Work Packet Review and Study Guide

Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

### MATH98 Intermediate Algebra Practice Test Form A

MATH98 Intermediate Algebra Practice Test Form A MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y - 4) - (y + ) = 3y 1) A)

### Geometry 21 Summer Work Packet Review and Study Guide

Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

### MATH 190 KHAN ACADEMY VIDEOS

MATH 10 KHAN ACADEMY VIDEOS MATTHEW AUTH 11 Order of operations 1 The Real Numbers (11) Example 11 Worked example: Order of operations (PEMDAS) 7 2 + (7 + 3 (5 2)) 4 2 12 Rational + Irrational Example

### = 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:

Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Order of Operations Expression Variable Coefficient

### P4 Polynomials and P5 Factoring Polynomials

P4 Polynomials and P5 Factoring Polynomials Professor Tim Busken Graduate T.A. Dynamical Systems Program Department of Mathematics San Diego State University June 22, 2011 Professor Tim Busken (Graduate

### Are you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*

Name: Date: Period: Are you ready for Algebra? Summer Packet *Required for all Students* The course prepares students for Pre Calculus and college math courses. In order to accomplish this, the course