POLYNOMIALS CHAPTER 2. (A) Main Concepts and Results

Size: px
Start display at page:

Download "POLYNOMIALS CHAPTER 2. (A) Main Concepts and Results"

Transcription

1 CHAPTER POLYNOMIALS (A) Main Concepts and Results Meaning of a Polynomial Degree of a polynomial Coefficients Monomials, Binomials etc. Constant, Linear, Quadratic Polynomials etc. Value of a polynomial for a given value of the variable Zeroes of a polynomial Remainder theorem Factor theorem Factorisation of a quadratic polynomial by splitting the middle term Factorisation of algebraic epressions by using the Factor theorem Algebraic identities ( + y) = + y + y ( y) = y + y y = ( + y) ( y) ( + a) ( + b) = + (a + b) + ab ( + y + z) = + y + z + y + yz + z ( + y) = + y + y + y = + y + y ( + y) ( y) = y + y y = y y ( y) 90504

2 4 EXEMPLAR PROBLEMS + y = ( + y) ( y + y ) y = ( y) ( + y + y ) + y + z yz = ( + y + z) ( + y + z y yz z) (B) Multiple Choice Questions Sample Question : If + k + 6 = ( + ) ( + ) for all, then the value of k is (A) (B) (C) 5 (D) Solution : Answer (C) EXERCISE. Write the correct answer in each of the following :. Which one of the following is a polynomial? (A) (B) (C) + (D). is a polynomial of degree + (A) (B) 0 (C) (D). Degree of the polynomial is (A) 4 (B) 5 (C) (D) 7 4. Degree of the zero polynomial is (A) 0 (B) (C) Any natural number (D) 5. If ( ) Not defined = +, then ( ) p p is equal to (A) 0 (B) (C) 4 (D) The value of the polynomial 5 4 +, when = is (A) 6 (B) 6 (C) (D) 90504

3 POLYNOMIALS 5 7. If p() = +, then p() + p( ) is equal to (A) (B) (C) 0 (D) 6 8. Zero of the zero polynomial is (A) 0 (B) (C) Any real number (D) Not defined 9. Zero of the polynomial p() = + 5 is (A) 5 (B) 5 (C) 0. One of the zeroes of the polynomial is (A) (B) (C). If is divided by +, the remainder is 5 (D) (D) (A) 0 (B) (C) 49 (D) 50. If + is a factor of the polynomial + k, then the value of k is (A) (B) 4 (C) (D). + is a factor of the polynomial (A) + + (B) (C) (D) One of the factors of (5 ) + ( + 5) is (A) 5 + (B) 5 (C) 5 (D) 0 5. The value of is (A) (B) 477 (C) 487 (D) The factorisation of is (A) ( + ) ( + ) (B) ( + ) ( + ) (C) ( + ) ( + 5) (D) ( ) ( ) 7. Which of the following is a factor of ( + y) ( + y )? (A) + y + y (B) + y y (C) y (D) y 8. The coefficient of in the epansion of ( + ) is (A) (B) 9 (C) 8 (D) 7 y 9. If y + = (, y 0 ), the value of y is

4 6 EXEMPLAR PROBLEMS (A) (B) (C) 0 (D) 0. If 49 b = 7 + 7, then the value of b is (A) 0 (B) (C). If a + b + c = 0, then a + b + c is equal to (A) 0 (B) abc (C) abc (D) abc (C) Short Answer Questions with Reasoning Sample Question : Write whether the following statements are True or False. Justify your answer. Solution : 5 + is a polynomial (ii) (D) is a polynomial, 0 False, because the eponent of the variable is not a whole number. (ii) True, because 6 + = 6 +, which is a polynomial. EXERCISE.. Which of the following epressions are polynomials? Justify your answer: 8 (ii) (iii) 5 (iv) (v) ( )( 4) (vi) + (vii) a a + 4 a 7 (viii)

5 POLYNOMIALS 7. Write whether the following statements are True or False. Justify your answer. (ii) A binomial can have atmost two terms Every polynomial is a binomial (iii) A binomial may have degree 5 (iv) Zero of a polynomial is always 0 (v) A polynomial cannot have more than one zero (vi) The degree of the sum of two polynomials each of degree 5 is always 5. (D) Short Answer Questions Sample Question : Check whether p() is a multiple of g() or not, where (ii) Solution : p() = +, g() = Check whether g() is a factor of p() or not, where p() = , g() = 4 p() will be a multiple of g() if g() divides p(). Now, g() = = 0 gives = Remainder = p = + = 8 + = Since remainder 0, so, p() is not a multiple of g(). (ii) g() = = 0 gives = 4 4 g() will be a factor of p() if p = 0 (Factor theorem) 4 Now, p =

6 8 EXEMPLAR PROBLEMS Since, 7 9 = = p = 0, so, g() is a factor of p(). 4 Sample Question : Find the value of a, if a is a factor of a + + a. Solution : Let p() = a + + a Since a is a factor of p(), so p(a) = 0. i.e., a a(a) + a + a = 0 a a + a + a = 0 a = Therefore, a = Sample Question : Without actually calculating the cubes, find the value of (ii)without finding the cubes, factorise ( y) + (y z) + (z ). Solution : We know that + y + z yz = ( + y + z) ( + y + z y yz z). If + y + z = 0, then + y + z yz = 0 or + y + z = yz. We have to find the value of = 48 + ( 0) + ( 8). Here, 48 + ( 0) + ( 8) = 0 So, 48 + ( 0) + ( 8) = 48 ( 0) ( 8) = (ii) Here, ( y) + (y z) + (z ) = 0 Therefore, ( y) + (y z) + (z ) = ( y) (y z) (z ). EXERCISE.. Classify the following polynomials as polynomials in one variable, two variables etc. + + (ii) y 5y (iii) y + yz + z (iv) y + y

7 POLYNOMIALS 9. Determine the degree of each of the following polynomials : (ii) 0 (iii) (iv) y ( y 4 ). For the polynomial , write 5 the degree of the polynomial (ii) the coefficient of (iii) the coefficient of 6 (iv) the constant term 4. Write the coefficient of in each of the following : π 6 + (ii) 5 (iii) ( ) ( 4) (iv) ( 5) ( + ) 5. Classify the following as a constant, linear, quadratic and cubic polynomials : + (ii) (iii) 5 t 7 (iv) 4 5y (v) (vi) + (vii) y y (viii) + + (i) t () 6. Give an eample of a polynomial, which is : monomial of degree (ii) binomial of degree 0 (iii) trinomial of degree 7. Find the value of the polynomial , when = and also when =. 8. If p() = 4 +, evaluate : p() p( ) + p 9. Find p(0), p(), p( ) for the following polynomials : p() = 0 4 (ii) p(y) = (y + ) (y ) 0. Verify whether the following are True or False : is a zero of 90504

8 0 EXEMPLAR PROBLEMS (ii) is a zero of + (iii) (iv) 4 5 is a zero of 4 5y 0 and are the zeroes of t t (v) is a zero of y + y 6. Find the zeroes of the polynomial in each of the following : p() = 4 (ii) g() = 6 (iii) q() = 7 (iv) h(y) = y. Find the zeroes of the polynomial : p() = ( ) ( + ). By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial : 4 + ; 4. By Remainder Theorem find the remainder, when p() is divided by g(), where p() = 4, g() = + (ii) p() = , g() = (iii) p() = 4 + 4, g() = (iv) p() = 6 + 4, g() = 5. Check whether p() is a multiple of g() or not : p() = 5 + 4, g() = (ii) p() = 4 + 5, g() = + 6. Show that : + is a factor of (ii) is a factor of Determine which of the following polynomials has a factor : (ii) Show that p is a factor of p 0 and also of p. 9. For what value of m is m + 6 divisible by +? 0. If + a is a factor of 5 4a + + a +, find a.. Find the value of m so that be a factor of m

9 POLYNOMIALS. If + is a factor of a + + 4a 9, find the value of a.. Factorise : (ii) (iii) 7 5 (iv) 84 r r 4. Factorise : (ii) (iii) (iv) + 5. Using suitable identity, evaluate the following: 0 (ii) 0 0 (iii) Factorise the following: (ii) 9y 66yz + z (iii) + 7. Factorise the following : 9 + (ii) Epand the following : (4a b + c) (ii) (a 5b c) (iii) ( + y z) 9. Factorise the following : 9 + 4y + 6z + y 6yz 4z (ii) 5 + 6y + 4z 40y + 6yz 0z (iii) 6 + 4y + 9z 6y yz + 4 z 0. If a + b + c = 9 and ab + bc + ca = 6, find a + b + c.. Epand the following : y (a b) (ii) +. Factorise the following : 64a a + 48a (iii)

10 EXEMPLAR PROBLEMS (ii) p p p. Find the following products : 4. Factorise : + y y + 4y 4 (ii) ( ) ( ) + 64 (ii) a b 5. Find the following product : ( y + z) (4 + y + 9z + y + yz 6z) 6. Factorise : a 8b 64c 4abc (ii) a + 8b 7c + 8 abc. 7. Without actually calculating the cubes, find the value of : Without finding the cubes, factorise ( y) + (y z) + (z ) 9. Find the value of + y y + 64, when + y = 4 (ii) 8y 6y 6, when = y + 6 (ii) (0.) (0.) + (0.) 40. Give possible epressions for the length and breadth of the rectangle whose area is given by 4a + 4a. (E) Long Answer Questions Sample Question : If + y = and y = 7, find the value of + y. Solution : + y = ( + y) ( y + y ) = ( + y) [( + y) y] = [ 7] = 6 =

11 POLYNOMIALS Alternative Solution : + y = ( + y) y ( + y) = 7 = [ 7] = 6 = 756 EXERCISE.4. If the polynomials az + 4z + z 4 and z 4z + a leave the same remainder when divided by z, find the value of a.. The polynomial p() = 4 + a + a 7 when divided by + leaves the remainder 9. Find the values of a. Also find the remainder when p() is divided by +.. If both and are factors of p r, show that p = r. 4. Without actual division, prove that is divisible by +. [Hint: Factorise + ] 5. Simplify ( 5y) ( + 5y). 6. Multiply + 4y + z + y + z yz by ( z + y). 7. If a, b, c are all non-zero and a + b + c = 0, prove that a b c + + =. bc ca ab 8. If a + b + c = 5 and ab + bc + ca = 0, then prove that a + b + c abc = Prove that (a + b + c) a b c = (a + b ) (b + c) (c + a)

ILLUSTRATIVE EXAMPLES

ILLUSTRATIVE EXAMPLES CHAPTER Points to Remember : POLYNOMIALS 7. A symbol having a fied numerical value is called a constant. For e.g. 9,,, etc.. A symbol which may take different numerical values is known as a variable. We

More information

Class IX Chapter 2 Polynomials Maths

Class IX Chapter 2 Polynomials Maths NCRTSOLUTIONS.BLOGSPOT.COM Class IX Chapter 2 Polynomials Maths Exercise 2.1 Question 1: Which of the following expressions are polynomials in one variable and which are No. It can be observed that the

More information

THE DISTRIBUTIVE LAW. Note: To avoid mistakes, include arrows above or below the terms that are being multiplied.

THE DISTRIBUTIVE LAW. Note: To avoid mistakes, include arrows above or below the terms that are being multiplied. THE DISTRIBUTIVE LAW ( ) When an equation of the form a b c is epanded, every term inside the bracket is multiplied by the number or pronumeral (letter), and the sign that is located outside the brackets.

More information

ISSUED BY KENDRIYA VIDYALAYA - DOWNLOADED FROM Chapter - 2. (Polynomials)

ISSUED BY KENDRIYA VIDYALAYA - DOWNLOADED FROM   Chapter - 2. (Polynomials) Chapter - 2 (Polynomials) Key Concepts Constants : A symbol having a fixed numerical value is called a constant. Example : 7, 3, -2, 3/7, etc. are all constants. Variables : A symbol which may be assigned

More information

Algebraic Expressions and Identities

Algebraic Expressions and Identities 9 Algebraic Epressions and Identities introduction In previous classes, you have studied the fundamental concepts of algebra, algebraic epressions and their addition and subtraction. In this chapter, we

More information

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

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

More information

TECHNIQUES IN FACTORISATION

TECHNIQUES IN FACTORISATION TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. METHOD: Epansion ( + )( 5) 15 Factorisation

More information

Algebraic Expressions

Algebraic Expressions Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly

More information

ALGEBRAIC EXPRESSIONS AND POLYNOMIALS

ALGEBRAIC EXPRESSIONS AND POLYNOMIALS MODULE - ic Epressions and Polynomials ALGEBRAIC EXPRESSIONS AND POLYNOMIALS So far, you had been using arithmetical numbers, which included natural numbers, whole numbers, fractional numbers, etc. and

More information

Algebra Final Exam Review Packet

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:

More information

Algebra. x + a 0. x 2 + a 1. are constants and a n. x n 1,..., a 0. x n, a n 1

Algebra. x + a 0. x 2 + a 1. are constants and a n. x n 1,..., a 0. x n, a n 1 2 Algebra Introduction We have studied about a particular type of algebraic expression, called polynomial, and the terminology related to it. Now, we shall study the factorisation of polynomials. In addition,

More information

Algebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background

Algebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked

More information

TEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor

TEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor POLYNOMIALS UNIT TEKS: A.10F Terms: Functions Equations Inequalities Linear Domain Factor Polynomials Monomial, Like Terms, binomials, leading coefficient, degree of polynomial, standard form, terms, Parent

More information

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

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

More information

CLASS IX MATHS CHAPTER REAL NUMBERS

CLASS IX MATHS CHAPTER REAL NUMBERS Previous knowledge question Ques. Define natural numbers? CLASS IX MATHS CHAPTER REAL NUMBERS counting numbers are known as natural numbers. Thus,,3,4,. etc. are natural numbers. Ques. Define whole numbers?

More information

1. A polynomial p(x) in one variable x is an algebraic expression in x of the form

1. A polynomial p(x) in one variable x is an algebraic expression in x of the form POLYNOMIALS Important Points 1. A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = a nx n +a n-1x n-1 + a 2x 2 +a 1x 1 +a 0x 0 where a 0, a 1, a 2 a n are constants

More information

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

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

More information

Eby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it

Eby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it Eby, MATH 010 Spring 017 Page 5 5.1 Eponents Parentheses are IMPORTANT!! Eponents only change what they! So if a is not inside parentheses, then it get raised to the power! Eample 1 4 b) 4 c) 4 ( ) d)

More information

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power. MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world

More information

Partial Fractions. Prerequisites: Solving simple equations; comparing coefficients; factorising simple quadratics and cubics; polynomial division.

Partial Fractions. Prerequisites: Solving simple equations; comparing coefficients; factorising simple quadratics and cubics; polynomial division. Prerequisites: olving simple equations; comparing coefficients; factorising simple quadratics and cubics; polynomial division. Maths Applications: Integration; graph sketching. Real-World Applications:

More information

Algebraic Expressions and Identities

Algebraic Expressions and Identities ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 Algebraic Expressions and Identities CHAPTER 9 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or

More information

Chapter 5: Exponents and Polynomials

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

More information

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

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

More information

Math 154 :: Elementary Algebra

Math 154 :: Elementary Algebra Math :: Elementar Algebra Section. Section. Section. Section. Section. Section. Math :: Elementar Algebra Section. Eponents. When multipling like-bases, ou can add the eponents to simplif the epression..

More information

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a

More information

Expanding brackets and factorising

Expanding brackets and factorising CHAPTER 8 Epanding brackets and factorising 8 CHAPTER Epanding brackets and factorising 8.1 Epanding brackets There are three rows. Each row has n students. The number of students is 3 n 3n. Two students

More information

6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property

6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property 6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.

More information

Unit 3 Factors & Products

Unit 3 Factors & Products 1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors

More information

A2T. Rational Expressions/Equations. Name: Teacher: Pd:

A2T. Rational Expressions/Equations. Name: Teacher: Pd: AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest

More information

Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms

Polynomial comes from poly- (meaning many) and -nomial (in this case meaning term)... so it says many terms Polynomials Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms Polynomials A polynomial looks like this: Term A number, a variable, or the

More information

27 Wyner Math 2 Spring 2019

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

More information

Polynomials and Polynomial Functions

Polynomials and Polynomial Functions Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial

More information

Multiplying Monomials

Multiplying Monomials 320 Chapter 5 Polynomials Eample 1 Multiplying Monomials Multiply the monomials. a. 13 2 y 7 215 3 y2 b. 1 3 4 y 3 21 2 6 yz 8 2 a. 13 2 y 7 215 3 y2 13 521 2 3 21y 7 y2 15 5 y 8 Group coefficients and

More information

Lesson 10.1 Polynomials

Lesson 10.1 Polynomials Lesson 10.1 Polynomials Objectives Classify polynomials. Use algebra tiles to add polynomials. Add and subtract polynomials. A contractor is buying paint to cover the interior of two cubical storage tanks.

More information

Common Core Algebra 2 Review Session 1

Common Core Algebra 2 Review Session 1 Common Core Algebra 2 Review Session 1 NAME Date 1. Which of the following is algebraically equivalent to the sum of 4x 2 8x + 7 and 3x 2 2x 5? (1) 7x 2 10x + 2 (2) 7x 2 6x 12 (3) 7x 4 10x 2 + 2 (4) 12x

More information

POLYNOMIALS. x + 1 x x 4 + x 3. x x 3 x 2. x x 2 + x. x + 1 x 1

POLYNOMIALS. x + 1 x x 4 + x 3. x x 3 x 2. x x 2 + x. x + 1 x 1 POLYNOMIALS A polynomial in x is an expression of the form p(x) = a 0 + a 1 x + a x +. + a n x n Where a 0, a 1, a. a n are real numbers and n is a non-negative integer and a n 0. A polynomial having only

More information

Polynomials. Title Page. Prior Knowledge: understanding of coefficients and terms

Polynomials. Title Page. Prior Knowledge: understanding of coefficients and terms Polynomials Title Page Subject: Polynomials Topic: Classifying Polynomials by Degree and Number of Terms. Also a review of Coefficients. Grade(s): 8th 10th grade Prior Knowledge: understanding of coefficients

More information

Factorisation CHAPTER Introduction

Factorisation CHAPTER Introduction FACTORISATION 217 Factorisation CHAPTER 14 14.1 Introduction 14.1.1 Factors of natural numbers You will remember what you learnt about factors in Class VI. Let us take a natural number, say 30, and write

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

More information

Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers

Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,

More information

5.1 The Language of Mathematics

5.1 The Language of Mathematics 5. The Language of Mathematics Prescribed Learning Outcomes (PLO s): Use mathematical terminology (variables, degree, number of terms, coefficients, constant terms) to describe polynomials. Identify different

More information

What is a constant? A Constant is a number representing a quantity or value that does not change.

What is a constant? A Constant is a number representing a quantity or value that does not change. Worksheet -: Algebraic Expressions What is a constant? A Constant is a number representing a quantity or value that does not change. What is a variable? A variable is a letter or symbol representing a

More information

Pre-Algebra Notes Unit 12: Polynomials and Sequences

Pre-Algebra Notes Unit 12: Polynomials and Sequences Pre-Algebra Notes Unit 1: Polynomials and Sequences Polynomials Syllabus Objective: (6.1) The student will write polynomials in standard form. Let s review a definition: monomial. A monomial is a number,

More information

Honours Advanced Algebra Unit 2: Polynomial Functions What s Your Identity? Learning Task (Task 8) Date: Period:

Honours Advanced Algebra Unit 2: Polynomial Functions What s Your Identity? Learning Task (Task 8) Date: Period: Honours Advanced Algebra Name: Unit : Polynomial Functions What s Your Identity? Learning Task (Task 8) Date: Period: Introduction Equivalent algebraic epressions, also called algebraic identities, give

More information

Polynomials and Factoring

Polynomials and Factoring 7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of

More information

Algebra 1 Mod 1 Review Worksheet I. Graphs Consider the graph below. Please do this worksheet in your notebook, not on this paper.

Algebra 1 Mod 1 Review Worksheet I. Graphs Consider the graph below. Please do this worksheet in your notebook, not on this paper. Algebra 1 Mod 1 Review Worksheet I. Graphs Consider the graph below Please do this worksheet in your notebook, not on this paper. A) For the solid line; calculate the average speed from: 1) 1:00 pm to

More information

not to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results

not to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division

More information

Honors Advanced Algebra Unit 2 Polynomial Operations September 14, 2016 Task 7: What s Your Identity?

Honors Advanced Algebra Unit 2 Polynomial Operations September 14, 2016 Task 7: What s Your Identity? Honors Advanced Algebra Name Unit Polynomial Operations September 14, 016 Task 7: What s Your Identity? MGSE9 1.A.APR.4 Prove polynomial identities and use them to describe numerical relationships. MGSE9

More information

CM2104: Computational Mathematics General Maths: 2. Algebra - Factorisation

CM2104: Computational Mathematics General Maths: 2. Algebra - Factorisation CM204: Computational Mathematics General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of simplifying algebraic expressions.

More information

Classifying Polynomials. Classifying Polynomials by Numbers of Terms

Classifying Polynomials. Classifying Polynomials by Numbers of Terms Lesson -2 Lesson -2 Classifying Polynomials BIG IDEA Polynomials are classifi ed by their number of terms and by their degree. Classifying Polynomials by Numbers of Terms Recall that a term can be a single

More information

Section 10-1: Laws of Exponents

Section 10-1: Laws of Exponents Section -: Laws of Eponents Learning Outcome Multiply: - ( ) = - - = = To multiply like bases, add eponents, and use common base. Rewrite answer with positive eponent. Learning Outcome Write the reciprocals

More information

EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n

EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m

More information

Composition of and the Transformation of Functions

Composition of and the Transformation of Functions 1 3 Specific Outcome Demonstrate an understanding of operations on, and compositions of, functions. Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of

More information

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 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

More information

Dr. Relja Vulanovic Professor of Mathematics Kent State University at Stark c 2008

Dr. Relja Vulanovic Professor of Mathematics Kent State University at Stark c 2008 MATH-LITERACY MANUAL Dr. Relja Vulanovic Professor of Mathematics Kent State University at Stark c 2008 2 Algebraic Epressions 2.1 Terms and Factors 29 2.2 Types of Algebraic Epressions 32 2.3 Transforming

More information

Properties of Real Numbers

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

More information

Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2

Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2 Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Factor each expression. 1. 3x 6y 2. a 2 b 2 3(x 2y) (a + b)(a b) Find each product. 3. (x 1)(x + 3) 4. (a + 1)(a 2 + 1) x 2 + 2x 3 a 3 + a 2 +

More information

Basic Algebra. Mathletics Instant Workbooks. 7(4x - y) = Copyright

Basic Algebra. Mathletics Instant Workbooks. 7(4x - y) = Copyright Basic Algebra Student Book - Series I- 7(4 - y) = Mathletics Instant Workbooks Copyright Student Book - Series I Contents Topics Topic - Addition and subtraction of like terms Topic 2 - Multiplication

More information

Mathematics Extension 2 HSC Examination Topic: Polynomials

Mathematics Extension 2 HSC Examination Topic: Polynomials by Topic 995 to 006 Polynomials Page Mathematics Etension Eamination Topic: Polynomials 06 06 05 05 c Two of the zeros of P() = + 59 8 + 0 are a + ib and a + ib, where a and b are real and b > 0. Find

More information

CHAPTER 1 POLYNOMIALS

CHAPTER 1 POLYNOMIALS 1 CHAPTER 1 POLYNOMIALS 1.1 Removing Nested Symbols of Grouping Simplify. 1. 4x + 3( x ) + 4( x + 1). ( ) 3x + 4 5 x 3 + x 3. 3 5( y 4) + 6 y ( y + 3) 4. 3 n ( n + 5) 4 ( n + 8) 5. ( x + 5) x + 3( x 6)

More information

5.1 Monomials. Algebra 2

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

More information

Divisibility Rules Algebra 9.0

Divisibility Rules Algebra 9.0 Name Period Divisibility Rules Algebra 9.0 A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following eercise: 1. Cross

More information

Algebra I Notes Unit Eleven: Polynomials

Algebra I Notes Unit Eleven: Polynomials Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Teacher Note: A nice way to illustrate operations with polynomials

More information

Pre-Algebra 2. Unit 9. Polynomials Name Period

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:

More information

Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

More information

Algebra I Lesson 6 Monomials and Polynomials (Grades 9-12) Instruction 6-1 Multiplying Polynomials

Algebra I Lesson 6 Monomials and Polynomials (Grades 9-12) Instruction 6-1 Multiplying Polynomials In algebra, we deal with different types of expressions. Grouping them helps us to learn rules and concepts easily. One group of expressions is called polynomials. In a polynomial, the powers are whole

More information

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

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

More information

Section 4.3: Quadratic Formula

Section 4.3: Quadratic Formula Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this

More information

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example: Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a

More information

Unit 13: Polynomials and Exponents

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:

More information

4.3 Division of Polynomials

4.3 Division of Polynomials 4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed

More information

MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline

MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 1-75 odd) A. Simplify fractions. B. Change mixed numbers

More information

SOLUTIONS. Math 130 Midterm Spring True-False: Circle T if the statement is always true. Otherwise circle F.

SOLUTIONS. Math 130 Midterm Spring True-False: Circle T if the statement is always true. Otherwise circle F. SOLUTIONS Math 13 Midterm Spring 16 Directions: True-False: Circle T if the statement is always true. Otherwise circle F. Partial-Credit: Please show all work and fully justify each step. No credit will

More information

Day 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions

Day 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions 1 Day : Section P-6 Rational Epressions; Section P-7 Equations Rational Epressions A rational epression (Fractions) is the quotient of two polynomials. The set of real numbers for which an algebraic epression

More information

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions

More information

A polynomial is an algebraic expression that has many terms connected by only the operations of +, -, and of variables.

A polynomial is an algebraic expression that has many terms connected by only the operations of +, -, and of variables. A polynomial is an algebraic expression that has many terms connected by only the operations of +, -, and of variables. 2x + 5 5 x 7x +19 5x 2-7x + 19 x 2 1 x + 2 2x 3 y 4 z x + 2 2x The terms are the

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t 4 3. 8s 4 + 4s 6 5s 4. 9m + 3 + m 3 + 5m 5. 4 3p 7 3p 4 1 z 1 + 4 6. ( ) 7. 6( x + ) 4 8. 3( h + 4) 3( h

More information

ANSWERS. CLASS: VIII TERM - 1 SUBJECT: Mathematics. Exercise: 1(A) Exercise: 1(B)

ANSWERS. CLASS: VIII TERM - 1 SUBJECT: Mathematics. Exercise: 1(A) Exercise: 1(B) ANSWERS CLASS: VIII TERM - 1 SUBJECT: Mathematics TOPIC: 1. Rational Numbers Exercise: 1(A) 1. Fill in the blanks: (i) -21/24 (ii) -4/7 < -4/11 (iii)16/19 (iv)11/13 and -11/13 (v) 0 2. Answer True or False:

More information

1. (a) 10, 5, 1, 0, 2, 3 (b) 12, 5, 3, 0, 3, 10 (c) 7, 3, 1, 2, 4, 10 (d) 31, 21, , 21, 41

1. (a) 10, 5, 1, 0, 2, 3 (b) 12, 5, 3, 0, 3, 10 (c) 7, 3, 1, 2, 4, 10 (d) 31, 21, , 21, 41 Practice Book UNIT 0 Equations Unit 0: Equations 0 Negative Numbers (a) 0 0 (b) 0 0 (c) 7 4 0 (d) 4 (a) 4 (b) 0 (c) 6 4 (d) 4 (a) < (b) < 4 (c) > 8 (d) 0 > 0 (a) 0 (b) 0 (c) July (d) 0 Arithmetic with

More information

1.6 Multiplying and Dividing Rational Expressions

1.6 Multiplying and Dividing Rational Expressions 1.6 Multiplying and Dividing Rational Epressions The game of badminton originated in England around 1870. Badminton is named after the Duke of Beaufort s home, Badminton House, where the game was first

More information

MATH Spring 2010 Topics per Section

MATH Spring 2010 Topics per Section MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line

More information

Basic ALGEBRA 2 SUMMER PACKET

Basic ALGEBRA 2 SUMMER PACKET Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout

More information

Ch. 9.3 Vertex to General Form. of a Parabola

Ch. 9.3 Vertex to General Form. of a Parabola Ch. 9.3 Verte to General Form Learning Intentions: of a Parabola Change a quadratic equation from verte to general form. Learn to square a binomial & factor perfectsquare epressions using rectangle diagrams.

More information

Controlling the Population

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

More information

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

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

More information

Algebra I Polynomials

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

More information

24. Find, describe, and correct the error below in determining the sum of the expressions:

24. Find, describe, and correct the error below in determining the sum of the expressions: SECONDARY 3 HONORS ~ Unit 2A Assignments SECTION 2.2 (page 69): Simplify each expression: 7. 8. 9. 10. 11. Given the binomials and, how would you find the product? 13. Is the product of two polynomials

More information

Bishop Kelley High School Summer Math Program Course: Algebra 2 A

Bishop Kelley High School Summer Math Program Course: Algebra 2 A 06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems

More information

Jakarta International School 8 th Grade AG1 Practice Test - BLACK

Jakarta International School 8 th Grade AG1 Practice Test - BLACK Jakarta International School 8 th Grade AG1 Practice Test - BLACK Polynomials and Quadratic Equations Name: Date: Grade: Standard Level Learning Goals - Green Understand and Operate with Polynomials Graph

More information

12x y (4) 2x y (4) 5x y is the same as

12x y (4) 2x y (4) 5x y is the same as Name: Unit #6 Review Quadratic Algebra Date: 1. When 6 is multiplied b the result is 0 1 () 9 1 () 9 1 () 1 0. When is multiplied b the result is 10 6 1 () 7 1 () 7 () 10 6. Written without negative eponents

More information

COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Summer Review of Algebra. Designed for the Student Entering Accelerated/Honors Analysis

COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Summer Review of Algebra. Designed for the Student Entering Accelerated/Honors Analysis COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Summer Review of Algebra Designed for the Student Entering Accelerated/Honors Analysis Please print this packet in its entirety to be completed for evaluation on

More information

Chapter 5 Rational Expressions

Chapter 5 Rational Expressions Worksheet 4 (5.1 Chapter 5 Rational Expressions 5.1 Simplifying Rational Expressions Summary 1: Definitions and General Properties of Rational Numbers and Rational Expressions A rational number can be

More information

POLYNOMIALS. Maths 4 th ESO José Jaime Noguera

POLYNOMIALS. Maths 4 th ESO José Jaime Noguera POLYNOMIALS Maths 4 th ESO José Jaime Noguera 1 Algebraic expressions Book, page 26 YOUR TURN: exercises 1, 2, 3. Exercise: Find the numerical value of the algebraic expression xy 2 8x + y, knowing that

More information

Polynomial and Synthetic Division

Polynomial and Synthetic Division Chapter Polynomial Functions. f y. Common function: y Transformation: Vertical stretch each y-value is multiplied by, then a vertical shift nine units upward f Horizontal shift of three units to the left,

More information

Algebra I Semester 2 Practice Exam DRAFT

Algebra I Semester 2 Practice Exam DRAFT Algebra I Semester Practice Eam 1. What is the -coordinate of the point of intersection for the two lines below? 6 7 y y 640 8 13 4 13. What is the y-coordinate of the point of intersection for the two

More information

Which of the following expressions are monomials?

Which of the following expressions are monomials? 9 1 Stud Guide Pages 382 387 Polnomials The epressions, 6, 5a 2, and 10cd 3 are eamples of monomials. A monomial is a number, a variable, or a product of numbers and variables. An eponents in a monomial

More information

Bishop Kelley High School Summer Math Program Course: Algebra 2 A

Bishop Kelley High School Summer Math Program Course: Algebra 2 A 015 016 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 16 pages of this packet provide eamples as to how to work some of the problems

More information

Graphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks).

Graphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks). Graphs of Polynomials: Polynomial functions of degree or higher are smooth and continuous. (No sharp corners or breaks). These are graphs of polynomials. These are NOT graphs of polynomials There is a

More information

Honors Algebra 2 Quarterly #3 Review

Honors Algebra 2 Quarterly #3 Review Name: Class: Date: ID: A Honors Algebra Quarterly #3 Review Mr. Barr Multiple Choice Identify the choice that best completes the statement or answers the question. Simplify the expression. 1. (3 + i) +

More information

Expressing a Rational Fraction as the sum of its Partial Fractions

Expressing a Rational Fraction as the sum of its Partial Fractions PARTIAL FRACTIONS Dear Reader An algebraic fraction or a rational fraction can be, often, expressed as the algebraic sum of relatively simpler fractions called partial fractions. The application of partial

More information