CCNY Math Review Chapter 1: Fundamentals

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1 CCNY Math Review Chapter 1: Fundamentals To navigate this document, click any button below or any chapter heading in the orange strip above. To move forward or backward between frames, click keyboard arrow keys or roll the mouse wheel. Introduction Section 1: Fundamental operations 1.0: The real number line 1.1: Adding and subtracting numbers 1.2: Multiplying and dividing numbers 1.3: Powers and monomials 1.4: Polynomials 1.5: Operations on polynomials 1.6: Exercises 1.7: The long division algorithm 1.8: Factoring numbers and polynomials 1.9: Factoring polynomials in one letter 1.10: Factoring quadratic polynomials 1.11: Factoring polynomials in several letters 1.12: Exercises 1.13: Quiz review Section 2: Fractions 2.1: Review of fractions 2.2: Reducing fractions 2.3: Multiplying/dividing fractions 2.4: Factoring a sum of products 2.5: Adding fractions 2.6: Adding polynomial fractions 2.7: Complex fractions 2.8: Exercises 2.9: Quiz Review Section 3: Powers 3.1: Rational and negative exponents 3.2: Roots and radicals 3.3: Square and nth root identities 3.4: Converting between fractions, radicals, and negative powers 3.5: Exercises 3.6: Quiz Review Section 4: Expressions, formulas, and functions 4.1: Parentheses in numerical expressions 4.2: The order of operations 4.3: Substituting numbers in formulas 4.4: Rewriting polynomial expressions 4.5: Substituting expressions for letters 4.6: Substituting letters for expressions 4.7: Rewriting rational expressions 4.8: When do you need parentheses? 4.9: How functions work 4.10: Function evaluation 4.11: Using parentheses with functions 4.12: Function composition 4.13: Exercises 4.14: Quiz Review All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review: Table of contents 8/09/16 Frame 1

2 CCNY Math Review Chapter 1: Fundamentals Section 5: Equations 5.1: Solving equations 5.2: Linear equations 5.3: Polynomial equations 5.4: Quadratic equations 5.5: Rational equations 5.6: Equations with radicals 5.7: Complex numbers solve quadratic equations 5.8: Hidden quadratic equations 5.9: Exercises 5.10: Quiz Review Section 6: Inequalities 6.1: Rewriting inequalities 6.2: Absolute value inequalities 6.3: Linear inequalities 6.4: Polynomial inequalities 6.5: Rational inequalities 6.6: Exercises 6.7: Quiz Review Section 7: Modeling real life problems 7.1: How precalculus can save the dolphins 7.2: Animation for a range of oil spill rates 7.3: How calculus can save the deer 7.4: Exercises 7.5: Quiz Review Section 8: The X,Y-coordinate plane 8.1: The coordinate plane 8.2: Finding point coordinates 8.3: Plotting a point 8.4: Distance and midpoint formulas 8.5: Graphs of equations 8.6: Finding points on graphs 8.7: Circles 8.8: Symmetry 8.9: Exercises 8.10: Quiz Review Section 9: Lines in the plane 9.1: Straight lines and their graphs 9.2: Lines and their slopes 9.3: Parallel and perpendicular lines 9.4: Equations of lines 9.5: Exercises 9.6: Quiz Review All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review: Table of contents 8/09/16 Frame 2

3 Welcome to CCNY Math Review Prof. Stanley Ocken Department of Mathematics The City College of New York Fall 2016 Introduction The goal of these notes is to help you manage the leap from high school math to college math. In high school math, a large part of the curriculum is devoted to setting up and solving real-life problems, but formal algebra instruction is not heavily emphasized. College is different. Virtually every topic in the STEM curriculum requires the use of algebra and formulas, tools that are used to attack complex problems that arise in science, technology, and engineering. The following Beamer slide presentation enables you, the student, to proceed step-by-step and at your own pace though these Math Review notes. To do so, use the arrow keys or the mouse to move from slide to slide, forwards or backwards. You may also click on the chapter titles at the top of any slide (or in the index at the left, accessible from the Adobe Acrobat Toolbar) to jump to different sections of this document. As you read through this material, please Read each section carefully, making sure to master the included Examples. Memorize and understand all included What is definitions and How to procedures. Work out the online homework exercises. Success in college math requires doing algebra carefully, thoughtfully, and (eventually) automatically. Your goal will often be to formulate real-life problems mathematically and then to work out the math. To do that, you will need to develop your ability to perform long, multi-step, and error-free algebraic calculations. In Babylonia, Egypt, and Greece, thousands of years ago, measurement and geometry were used to solve real-world problems. But it soon became apparent that the clarity of mind honed by the study of mathematics was of fundamental value to society in other ways as well. Indeed, in Plato s Republic, a description of the ideal state, future leaders were required to study mathematics for ten years before going on to more practical affairs of ethics and government. In Book VII, Socrates asserts: In every man there is an eye of the mind which, when by other pursuits lost and dimmed, is, by the study of mathematics, purified and re-illuminated. This eye is more precious by far than ten thousand bodily eyes, for by it alone can truth be seen. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review: Introduction 8/09/16 Frame 3

4 Algebra is the grammar of mathematics. You need to master this grammar if you wish to solve math and science problems. When you write an essay you assemble letters to make words; words and punctuation to make phrases; phrases to make sentences; sentences to express ideas. When you write mathematics you assemble numbers, letters, and punctuation to make expressions such as 30x 2 (x ); expressions and an equals sign to make statements such as (x + 2)(x 2) = x 2 4; and statements to solve a problem. The grammar rules for math are at least as strict as those for language. Numbers are written with digits 0 through 9. Letters are special symbols. Sometimes a letter is a variable, sometimes it is an unknown, sometimes it is a placeholder for a number, and sometimes it is just a letter! Expressions are the phrases of mathematics. They are built from letters, numbers, and the following punctuation marks: Parentheses: ( ) Operators: + (add) (subtract) (multiply) A Fraction lines: is A divided by B. B Power notation: 5 3 is The elementary school symbols (for division) and (for multiplication) are seldom used in college-level math textbooks. Identities such as E = F, where E and F are expressions, are the sentences of mathematics. They assert that each expression can be obtained from the other by applying the laws of algebra. Exactly what this means will be explained in detail in these notes. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review: Introduction 8/09/16 Frame 4

5 Section 1.1: Fundamental operations 1.1.0: The real number line 1.1.1: Adding and subtracting numbers 1.1.2: Multiplying and dividing numbers 1.1.3: Powers and monomials 1.1.4: Polynomials 1.1.5: Adding, subtracting, and multiplying polynomials 1.1.6: Exercises 1.1.7: The long division algorithm 1.1.8: Factoring numbers and polynomials 1.1.9: Factoring polynomials in one letter : Factoring quadratic polynomials : Factoring polynomials in several letters : Exercises : Quiz review All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 5

6 1.1.0 The real number line Numbers are used for counting and measuring. We count a collection of objects by using natural numbers 1, 2, 3,,, We measure the size or extent of quantities such as length, area, and volume by using measuring numbers, which include 0, 1, 2, 3,,, and all the in-between numbers such as , 0.4, 3/4, 7/8 1, 9, 2, π that could be the length of a stick. These numbers are fine for describing sizes. But quantities such as temperature are expressed as numbers that are above or below 0. Real numbers consist of two versions of measuring numbers: positive numbers, to the right of 0 and shown in blue below, and negative numbers, to the left of 0 and shown in red. Temperature, velocity, and the slope of a line can be positive, 0, or negative. However, length, area, and volume can only be positive. Error warning: There is no such thing as negative length, negative area, or negative volume. The line below is a model of the real number line. The integers are the dots..., 3, 2, 1, 0, 1, 2, 3,... The natural numbers are 1, 2, 3,... Every point on the line is a real number. Think of the picture as showing a horizontal thermometer. Temperatures above 0 are positive, while temperatures below 0 are negative. Positive and negative real numbers The real number line looks like a ruler with reference point 0. The part from 0 to 1 is called a unit. 3 (negative 3) is the number 3 units to the left of (positive 3), usually written as 3, is 3 units to the right of 0. If x and y are real numbers, x < y means: x is to the left of y. The absolute value x of a number x is its distance from 0. Thus 3 and 3 are both equal to All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 6

7 Classifying real numbers Counting numbers, usually called natural numbers, can be used to measure things: A stick might be 7 feet long and weigh 17 pounds. But most quantities, including length, mass, and duration, require in-between measurements. Things that are measured can be split into pieces. When you split a length 1 stick into 7 equal parts, you get 7 small sticks, each with length 1. When you glue 7 together 17 such small sticks, you get a stick with length 17. In this fraction, the denominator 7 tells you 7 the number of equal parts into which the unit was split, while the numerator 17 tells you how many of these parts were glued together to form a longer stick. All such numbers (stick lengths) are called rational numbers. Most stick lengths are not rational: they are called irrational numbers. Two examples: If a square s side length is 1 meter, then the length of its diagonal is D = 2 meters. Although D can t be written as a fraction, it satisfies a simple equation: D 2 = 2. If a circle s diameter is 1 meter, its circumference is π meters. This number is called transcendental: it doesn t satisfy any polynomial equation whatsoever. A rational number is a decimal number if its denominator is a power of 10. For example, = = Numbers of this sort are terminating decimals: they can be written with a finite number of digits to to the right of the decimal point. Every terminating decimal is a rational number. However, most rational numbers are not terminating decimals. For example, 1 can be written only as a 3 non-terminating decimal This symbol looks suspicious: always beware of... To justify this notation we need the theory of infinite series, which is encountered in second or third semester calculus. Every real number is approximately equal to (written ) decimal numbers. For example, while π Example: Rewrite as a rational number. Answer: = = All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 7

8 Intervals on the real number line Algebra is used to measure distance on the number line. How to find the distance between real numbers The distance between real numbers a and b on the number line equals the right number minus the left number. Below, the distance between 3 and 5 is 5 ( 3) = = The distance between points is always a positive number: if b is to the right of a, then b > a and b a is a positive number. Error warning: There is no such thing as negative distance. Comparing numbers. Suppose you want to compare the positions of the numbers 3 and 5 on the number line. The following are different ways of saying the same thing: is less than 5; 5 is greater than 3; 3 < 5; 5 > 3; 3 is to the left of 5; 5 is to the right of 3. The symbols and are read less than or equal to and greater than or equal to. x 3 means: x = 3 or x < 3. x 3 means: x = 3 or x > 3. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 8

9 What is an interval? An interval is a connected piece of the number line. Example: Describe in three different ways the interval consisting of all real numbers to the right of 3 and also to the left of (or equal to) 5. Solution: inequality notation: 3 < x 5; interval notation: ( 3, 5]; a number line graph: is drawn as a hollow dot, to show that it is not included in (is missing from) the interval. 5 is drawn as a solid dot, to show that it is included in the interval. Example: There are four intervals with left endpoint 3 and right endpoint 5. Write the four intervals with left endpoint 3 and right endpoint 5 using interval notation, inequality notation, and a graph. Solution: ( 3, 5) is an open interval, written in inequality form as 3 < x < [ 3, 5] is a closed interval, written in inequality form as 3 x ( 3, 5] is a half open interval, written in inequality form as 3 < x [ 3, 5) is a half open interval, written in inequality form as 3 x < All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 9

10 Infinite intervals on the real line The symbol or +, for (positive) infinity, is used when an interval contains all numbers to the right of a certain point. Example: Use three kinds of notation to describe all numbers to the right of 3. Solution: inequality notation 3 < x; interval notation: ( 3, ); a number line graph: The symbol, for negative infinity, is used when an interval contains all numbers to the left of a certain point. Example: Use three kinds of notation to describe all number to the left of and including 3. Solution: inequality notation: x 3; interval notation: (, 3] ; a number line graph: All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 10

11 Absolute value and distance What is absolute value? The absolute value of a real number a, written a, is the positive number defined by a = a if a 0 and a = a if a 0. { a if a 0 This is often written as a =. a if a 0 The length of the line segment joining points a and b is the distance between them. The distance between points a and b is the absolute value of their difference, in either order: a b = b a. In math classes (unlike in physics), distance is a number expressed without a specific unit of length. Here are some examples: 38 = 38 = = = 0. There s nothing special about 38. The basic rule x = x is true for all real x. In particular, let x = b a. Then x = (b a) = b + a = a b and so x = a b. Since x = x, it follows that a b = b a. For example, 8 3 and 3 8 = 5 are both equal to 5. Example: Use three methods to find the distance between 3 and 7. Solution: Since 3 is to the right of 7, the distance is 3 ( 7) = = 4. The distance is the absolute value of the first number minus the second number: 3 ( 7) = = 4 = 4. The distance is the absolute value of the second number minus the first number: 7 ( 3) = = 4 = 4. We seem to have demonstrated the following: Distance and absolute value If a and b are real numbers, then a b = b a is the distance from a to b on the number line. This follows from the fact that the distance between two numbers represented as points on the number line is the right number minus the left number. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 11

12 There are two possibilities: If b is to the right of (or equal to) a, then b a 0: the distance between a and b is b a = b a, or If a is to the right of (or equal to) b, then a b 0: the distance between a and b is a b = a b. In the above statement, take b = 0. Then a b = a 0 = a is the distance between a and 0 on the number line. That s an important fact in itself: Distance from 0 and absolute value The absolute value of a real number is its distance from 0 on the number line. Absolute value inequalities are treated in Section 5.3?????, but watch out: Error warning: a + b = a + b is false. The above formula if a and b have the same sign but is incorrect if a and b have opposite signs. So it s wrong just half the time! But the laws of algebra must be correct all the time! All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 12

13 1.1.1 Adding and subtracting real numbers How to add real numbers To add numbers with the same sign: Add their absolute values and give the answer that sign. To add numbers with opposite signs: Subtract the smaller absolute value from the larger, then give the result the sign of the number with the larger absolute value. We often use a letter such as A or B to stand for a number. How to subtract real numbers A B = A + ( B) Visualize the rules: To add 5 to a number, move 5 right from the number. To add 5 to a number, move 5 left from the number. To subtract 5 from a number, move 5 left from the number. To subtract 5 from a number, move 5 right from the number. Sums and differences of real numbers The sum of A and B is A + B. The difference of A and B is A B. Example 1: Find each sum: = = ( 4) = (10 + 4) = = (10 4) = = +(10 4) = 6 Example 2: Find each difference: 10 4 = 6 10 ( 4) = 10 + (+4) = ( 4) = = (10 4) = 6 4 ( 10) = = +(10 4) = 6 Numbers and their opposites Let A be any real number. The opposite of A, written A, is the reflection of A across 0 on the number line. A + A = A + A = 0 Example: The opposite of 3 is 3, while the opposite of 3, written ( 3), is 3. A number and its opposite are equally far from 0, but are on opposite sides of 0. Check that = = 0. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 13

14 1.1.2 Multiplying and dividing numbers Real number multiplication When you multiply numbers A and B, the result is their product, written as AB or A B, and only occasionally as A B. The order of multiplication doesn t matter: A B = B A. How to multiply real numbers If A = 0 or B = 0 then AB = 0. If neither A nor B is 0 then: To multiply numbers with the same sign, multiply their absolute values and give the result a + sign. To multiply numbers with different signs,multiply their absolute values and give the result a sign. Example 3: Find each product: Same signs: 3 5 = 3 ( 5) = 3 5 = 15 Opposite signs: 3 5 = 3 5 = (3 5) = 15 Multiplying a real number A by 1 The product 1 A is equal to A, the opposite of A. Reminder: A B means A is not equal to B. Real number division When you divide A by B, the result is the quotient of A by B, written as A or A/B, and only B occasionally as A B. The order of division matters: A B B A. If B 0 then A B = C means: B C = A. Error warning: division by 0 doesn t make sense. Here s why: 30 0 = C means 0 C = 30. But this is impossible, since 0 times anything is 0, not 30. How to divide real numbers If A = 0 and B 0 then A B = 0. If B = 0, then A B is undefined. If neither A nor B is zero, then: To divide numbers with the same sign, divide their absolute values and give the answer a + sign. To divide numbers with opposite signs, divide their absolute values and give the answer a sign. Example 4: Find each quotient: Same signs: 20 4 = 5 and 20 4 = 5 Opposite signs: 20 4 = 5 and 20 4 = 5 All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 14

15 1.1.3 Powers and monomials For any number A, the 4 th power of A, written A 4, is the product AAAA of four A s. For any integer n 1, A n is the product with n factors equal to A. A n is called the n th power of A. In this power, the base is A and the exponent is n. How to work with powers of numbers Suppose A is any real number and m and n are non-negative integers 0, 1, 2, 3, 4, is undefined; A 0 = 1 if A 0 A 1 = A To multiply powers of the same base, add exponents: A m A n = A m+n To multiply the same powers of different bases, multiply bases: A m B m = (AB) m To raise a power to a power, multiply exponents: (A m ) n = A mn Example 5: Rewrite using the above power rules: 12 0 = 1; 0 12 = 0; (4 2 ) 3 = = = = 4 5 ; = (3 4) 2 = 12 2 Algebraic expressions use letters such as A, B, x, y. In these notes, a letter is called a variable only if it is the input to a function. Definition of monomial A monomial is an integer, a product of powers of distinct letters, or an integer times such a product. The monomial s coefficient is 1 if the monomial is a product of powers of letters; the integer, otherwise. Its letter part is the product of powers of letters. A monomial is called constant if it has no letter part. Typical monomials are 0, 3, x, 7xy, 5x 3 y 3, 12xyz 2, 7yx, and 3y 10 x 18 z 3. Note that 7xy = 7yx because (as is the case with numbers), the order of multiplication of letters doesn t matter. We say that each monomial can be rewritten as the other. Rewrite a product of numbers and letters as a monomial Reorder the product so that all numbers are to the left of all letters. Multiply the numbers to get the coefficient of the monomial. Reorder the letter part so that powers of the same letter are adjacent. Multiply powers of the same letter by adding exponents. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 15

16 Since the product of monomials is a product of numbers and letters, we know How to multiply monomials Rewrite their product as a monomial. Example 6: Rewrite each product as a monomial. 7x 2 2x 3 = 7 2 x 2 x 3 = 7 2x 2+3 = 14x 5 7x 3 2y 3 x 2 y 4 = 7 2x 3 x 2 y 3 y 4 = 14x 5 y 7 x 3 y 3 x 2 y 4 = x 3 x 2 y 3 y 4 = x 5 y 7 7xyxyz = 7xxyyz = 7x 2 y 2 z 8x 3 y 3 x 80 = 8x 3 x 80 y 3 = 8x 3+80 y 3 = 8x 83 y 3 More about monomials The degree of a monomial is the sum of the exponents in its letter part. The degree of a constant monomial c = cx 0 is 0 if c 0 but is undefined if c = 0. Like terms are monomials that can be rewritten with identical letter parts. How to add monomials To add like terms add their coefficients. Error warning: Unlike terms can t be added. A sum of unlike terms should be left as is! Example: 3yx 2 z + 6x 2 yz = 3x 2 yz + 6x 2 yz = (3 + 6)x 2 yz = 9x 2 yz Example 7: Rewrite each sum as a monomial. 3x 2 y + 6x 2 y = (3 + 6)x 2 y = 9x 2 y 5x 2 7x 2 = 5x 2 + 7x 2 = (5 + 7)x 2 = 2x 2 Faster: 5x 2 7x 2 = (5 7)x 2 = 2x 2 7xyx + 8yx 2 + 9x 2 y = 7x 2 y + 8x 2 y + 9x 2 y = ( )x 2 y = 24x 2 y Exercises: Rewrite each product as a monomial a) x 3 x 8 x 6 b) 3x 3 x 8 c) 3x 3 5x 8 d) ( 2x 3 )x( 5x 7 ) e) 3t 3 3t 8 Please answer all the above questions before you check your answers below. For example, 7x 2 y 2 x 3 and 9xy 2 x 4 are like terms since both can be rewritten with letter part x 5 y 2 : 7x 2 y 2 x 3 = 7x 5 y 2 and 9xy 2 x 4 = 9x 5 y 2. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 16

17 Since the product of monomials is a product of numbers and letters, we know How to multiply monomials Rewrite their product as a monomial. Example 6: Rewrite each product as a monomial. 7x 2 2x 3 = 7 2 x 2 x 3 = 7 2x 2+3 = 14x 5 7x 3 2y 3 x 2 y 4 = 7 2x 3 x 2 y 3 y 4 = 14x 5 y 7 x 3 y 3 x 2 y 4 = x 3 x 2 y 3 y 4 = x 5 y 7 7xyxyz = 7xxyyz = 7x 2 y 2 z 8x 3 y 3 x 80 = 8x 3 x 80 y 3 = 8x 3+80 y 3 = 8x 83 y 3 More about monomials The degree of a monomial is the sum of the exponents in its letter part. The degree of a constant monomial c = cx 0 is 0 if c 0 but is undefined if c = 0. Like terms are monomials that can be rewritten with identical letter parts. For example, 7x 2 y 2 x 3 and 9xy 2 x 4 are like terms since both can be rewritten with letter part x 5 y 2 : 7x 2 y 2 x 3 = 7x 5 y 2 and 9xy 2 x 4 = 9x 5 y 2. How to add monomials To add like terms add their coefficients. Error warning: Unlike terms can t be added. A sum of unlike terms should be left as is! Example: 3yx 2 z + 6x 2 yz = 3x 2 yz + 6x 2 yz = (3 + 6)x 2 yz = 9x 2 yz Example 7: Rewrite each sum as a monomial. 3x 2 y + 6x 2 y = (3 + 6)x 2 y = 9x 2 y 5x 2 7x 2 = 5x 2 + 7x 2 = (5 + 7)x 2 = 2x 2 Faster: 5x 2 7x 2 = (5 7)x 2 = 2x 2 7xyx + 8yx 2 + 9x 2 y = 7x 2 y + 8x 2 y + 9x 2 y = ( )x 2 y = 24x 2 y Exercises: Rewrite each product as a monomial a) x 3 x 8 x 6 b) 3x 3 x 8 c) 3x 3 5x 8 d) ( 2x 3 )x( 5x 7 ) e) 3t 3 3t 8 Please answer all the above questions before you check your answers below. a) x 3 x 8 x 6 = x 17 b) 3x 3 x 8 = 3x 11 c) 3x 3 5x 8 = 15x 11 d) 2x 3 x( 5x 7 ) = 10x 11 e) 3t 3 3t 8 = 9t 11 All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 16

18 1.1.4 Polynomials Definition of polynomial A polynomial is a sum of monomials with no like terms. How to rewrite a sum of products as a polynomial by collecting like terms Rewrite each product as a monomial with letters in alphabetical order. Reorder the sum so that like terms are adjacent. Add like terms. Since a sum of polynomials is a sum of products, the above instructions explain how to add polynomials. It is customary to order the terms of a polynomial in one letter so that the exponents of the letters powers decrease from left to right. Polynomials in one letter The polynomial s leading term is the power with the highest exponent. leading coefficient is the coefficient of the leading term. polynomial in more than one letter. These are discussed in computer science courses. Example 8: Rewrite each sum as a polynomial: 5x 2 2xy + 7x 2 + 8xy = 5x 2 + 7x 2 2xy + 8xy = 12x 2 + 6xy 3 5x 2 + x 8 + 7x x 8 = 4x 8 + 1x 8 5x 2 + 7x = (4 + 1)x 8 + ( 5 + 7)x 2 12 = 5x 8 + 2x 2 12 Exercise: Rewrite each polynomial with decreasing exponents: a) x x 3 b) 3x 2x 2 4x 8x 3 + 3x 2 c) t 3t 5 6t 2 + t 3 d) x 3 + x 2 + 4x 3 + 5x 2 + x 4 x 6 Please answer all the above questions before you check your answers below. There are several schemes for ordering the terms of a All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 17

19 1.1.4 Polynomials Definition of polynomial A polynomial is a sum of monomials with no like terms. How to rewrite a sum of products as a polynomial by collecting like terms Rewrite each product as a monomial with letters in alphabetical order. Reorder the sum so that like terms are adjacent. Add like terms. Since a sum of polynomials is a sum of products, the above instructions explain how to add polynomials. It is customary to order the terms of a polynomial in one letter so that the exponents of the letters powers decrease from left to right. Polynomials in one letter The polynomial s leading term is the power with the highest exponent. leading coefficient is the coefficient of the leading term. There are several schemes for ordering the terms of a polynomial in more than one letter. These are discussed in computer science courses. Example 8: Rewrite each sum as a polynomial: 5x 2 2xy + 7x 2 + 8xy = 5x 2 + 7x 2 2xy + 8xy = 12x 2 + 6xy 3 5x 2 + x 8 + 7x x 8 = 4x 8 + 1x 8 5x 2 + 7x = (4 + 1)x 8 + ( 5 + 7)x 2 12 = 5x 8 + 2x 2 12 Exercise: Rewrite each polynomial with decreasing exponents: a) x x 3 b) 3x 2x 2 4x 8x 3 + 3x 2 c) t 3t 5 6t 2 + t 3 d) x 3 + x 2 + 4x 3 + 5x 2 + x 4 x 6 Please answer all the above questions before you check your answers below. a) x 3 + x b) 8x 3 + x 2 x c) t 3 6t 2 2t 5 d) x 6 + x 4 + 5x 3 + 6x 2 All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 17

20 1.1.5 Subtracting and multiplying polynomials How to distribute a minus sign across a sum of terms Rewrite the sum with each term multiplied by 1. Example 9: Rewrite each expression without parentheses: (4 x + y 7z) = 4 + x y + 7z x 3 (2y 3z + 3b 4) = x 3 2y + 3z 3b + 4. Until now, uppercase letters represented real numbers. From now on, they can also stand for monomials or polynomials. How to do polynomial subtraction P (Q) Remove parentheses from Q by distributing the minus sign. Rewrite the result as a polynomial by collecting like terms. Example 10: Rewrite the difference as a polynomial: (x 2 + 2x) (2x 2 3x)= x 2 + 2x 2x 2 + 3x = x 2 + 5x Error warning: Make sure to distribute the minus sign when subtracting: (x 2 + 2x) (2x 2 3x) = x 2 + 2x 2x 2 3x = wrong answer! Error warning: The above examples show how, not when, to distribute the minus sign. For example, (a + 2) 2 is not equal to ( a 2) 2. This will be explained in the section on order of operations. How to multiply a monomial by a polynomial Multiply the monomial by each term of the polynomial and add the resulting products. This follows from the distributive law A(B + C + D +...) = AB + AC + AD +..., which will be discussed in detail later on. In this case, A is the monomial, which may be just a constant. Example 11: Rewrite each product as a polynomial. 3(5x 3x 2 + 1) = 3 5x + 3 ( 3x 2 ) = 15x 9x = 9x x + 3 (2x 3 )(x 2 5x + 4) = 2x 3 (x 2 ) 2x 3 (5x) + 2x 3 (4) = 2x 5 10x 4 + 8x 3 ( 2x 2 y)(x 2 y 5x + 4y 2 ) = ( 2x 2 y)(x 2 y) + ( 2x 2 y)( 5x) + ( 2x 2 y)(4y 2 ) = 2x 4 y x 3 y 8x 2 y 3. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 18

21 How to multiply polynomials P and Q Multiply each term of P by each term of Q. Add the resulting products. Rewrite the result as a polynomial by collecting like terms. Example 12: Rewrite each blue expression as a polynomial. x 2 (x + x 3 ) =? Let A = x 2, B = x, C = x 3. Then A(B + C) = AB + AC = x 2 x + x 2 x 3 = x 3 + x 5 (3x + 2)(2x + 5) =? Let A = 3x, B = 2, C = 2x.D = 5. Then (A + B)(C + D) = AC + AD + BC + BD = 3x 2x + 3x x = 6x x + 4x + 10= 6x x + 10 (3x )(3x 4 10) =? Let A = 3x 4, B = 10. Then (A + B)(A B) = A 2 B 2 = (3x 4 ) = 9x (2x + 3)(x 2 5x + 4) =? You don t need to write out the formula (A + B)(C + D + E) = AC + AD + AE + BC + BD + BE. Instead, use any one of the solution formats below. Method 1: Use the distributive law: (2x + 3)(x 2 5x + 4) = 2x (x 2 5x + 4) : 2x 3 10x 2 + 8x +3 (x 2 5x + 4) : 3x 2 15x + 12 Add the results: = 2x 3 10x 2 + 8x + 3x 2 15x + 12 Reorder: = 2x 3 10x 2 + 3x 2 + 8x 15x + 12 Add like terms: = 2x 3 7x 2 7x + 12 Method 2: Use the chart below. Each blue term is the product of the terms at the top of its column and the left of its row. x 2 5x +4 2x 2x 3 10x 2 +8x +3 3x 2 15x +12 Add and collect the blue terms: 2x 3 7x 2 7x + 12 Method 3: Imitate the decimal multiplication procedure: x 2 5x +4 2x +3 3x 2 15x +12 2x 3 10x 2 +8x 2x 3 7x 2 7x +12 All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 19

22 1.1.6 Exercises: multiplying polynomials Multiply out and rewrite each product as a polynomial. Degrees of the monomials should decrease from left to right. If two monomials have the same degree, place them in alphabetical order. For example x 2 y comes before y 3 because, in the dictionary, xxy precedes yyy. a) x(2 + x) b) x 6 (x 2 9) c) 3(x + 2)(x + 1) d) (3x 2)(x 2 + x 3 ) e) (2 x)(2 x 2 )(3x) f) (x 1)(x 2 + x + 1) g) (3x + 4)(x 2 2x + 1) h) (2 x 2 )(x 3 + 3x) i) (x 2 + x + 1)(x 2 + x + 1) j) (x 3y)(x + 3y) k) (x y)(x 2y)(x 3y) l) (x y) 2 m) (x y)(x + y) 2 n) (x + 3) 3 o) (2x 3) 3 p) (x + y + 2) 2 q) (x 2 + 2x + 3) 2 Please make sure that you have answered all the questions before you check your answers below. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 20

23 1.1.6 Exercises: multiplying polynomials Multiply out and rewrite each product as a polynomial. Degrees of the monomials should decrease from left to right. If two monomials have the same degree, place them in alphabetical order. For example x 2 y comes before y 3 because, in the dictionary, xxy precedes yyy. a) x(2 + x) b) x 6 (x 2 9) c) 3(x + 2)(x + 1) d) (3x 2)(x 2 + x 3 ) e) (2 x)(2 x 2 )(3x) f) (x 1)(x 2 + x + 1) g) (3x + 4)(x 2 2x + 1) h) (2 x 2 )(x 3 + 3x) i) (x 2 + x + 1)(x 2 + x + 1) j) (x 3y)(x + 3y) k) (x y)(x 2y)(x 3y) l) (x y) 2 m) (x y)(x + y) 2 n) (x + 3) 3 o) (2x 3) 3 p) (x + y + 2) 2 q) (x 2 + 2x + 3) 2 Please make sure that you have answered all the questions before you check your answers below. a) x 2 + 2x b) x 8 9x 6 c) 3x 2 + 9x + 6 d) 3x 4 + x 3 2x 2 e) 3x 4 6x 3 6x x f) x 3 1 g) 3x 3 2x 2 5x + 4 h) x 5 x 3 + 6x i) x 4 + 2x 3 + 3x 2 + 2x + 1 j) x 2 9y 2 k) x 3 6x 2 y + 11xy 2 6y 3 l) x 2 2xy + y 2 m) x 3 + x 2 y xy 2 y 3 n) x 3 + 9x x + 27 o) 8x 3 36x x 27 p) x 2 + y 2 + 2xy + 4x + 4y + 4 q) x 4 + 4x x x + 9 All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 20

24 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Click slowly through the following demonstration. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

25 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Click slowly through the following demonstration. x + 2 / x 3 + 3x 2 + 3x + 7 All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

26 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Click slowly through the following demonstration. x 2 x into x 3 is x 2. x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 Multiply x 2 by x + 2 All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

27 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Subtract x 3 + 2x 2 from the dividend to get x 2 + 3x + 7, which we call Diff. Click slowly through the following demonstration. x 2 x into x 3 is x 2. x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 Multiply x 2 by x + 2 x 2 + 3x + 7 Subtract to get Diff = x 2 + 3x + 7. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

28 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Subtract x 3 + 2x 2 from the dividend to get x 2 + 3x + 7, which we call Diff. If degree of Diff is less than degree of the divisor, Diff is the remainder: Stop. If not, repeat the above process using Diff as the new dividend. Click slowly through the following demonstration. x 2 + x x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 x 2 + 3x + 7 x 2 + 2x x into x 3 is x 2. Multiply x 2 by x + 2 Subtract to get Diff = x 2 + 3x + 7. x into x 2 is x. Multiply x by x + 2 All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

29 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Subtract x 3 + 2x 2 from the dividend to get x 2 + 3x + 7, which we call Diff. If degree of Diff is less than degree of the divisor, Diff is the remainder: Stop. If not, repeat the above process using Diff as the new dividend. Click slowly through the following demonstration. x 2 + x x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 x 2 + 3x + 7 x 2 + 2x x + 7 x into x 3 is x 2. Multiply x 2 by x + 2 Subtract to get Diff = x 2 + 3x + 7. x into x 2 is x. Multiply x by x + 2 Subtract to get Diff = x + 7. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

30 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Subtract x 3 + 2x 2 from the dividend to get x 2 + 3x + 7, which we call Diff. If degree of Diff is less than degree of the divisor, Diff is the remainder: Stop. If not, repeat the above process using Diff as the new dividend. Click slowly through the following demonstration. x 2 + x +1 x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 x 2 + 3x + 7 x 2 + 2x x + 7 x x into x 3 is x 2. Multiply x 2 by x + 2 Subtract to get Diff = x 2 + 3x + 7. x into x 2 is x. Multiply x by x + 2 Subtract to get Diff = x + 7. x into x is 1. Multiply 1 by x + 2 Subtract to get Diff = 5. At this point we stop because the degree 0 of 5 is less than the degree 1 of the divisor x + 2. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

31 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Subtract x 3 + 2x 2 from the dividend to get x 2 + 3x + 7, which we call Diff. If degree of Diff is less than degree of the divisor, Diff is the remainder: Stop. If not, repeat the above process using Diff as the new dividend. Click slowly through the following demonstration. x 2 + x +1 x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 x 2 + 3x + 7 x 2 + 2x x + 7 x x into x 3 is x 2. Multiply x 2 by x + 2 Subtract to get Diff = x 2 + 3x + 7. x into x 2 is x. Multiply x by x + 2 Subtract to get Diff = x + 7. x into x is 1. Multiply 1 by x + 2 Subtract to get Diff = 5. At this point we stop because the degree 0 of 5 is less than the degree 1 of the divisor x + 2. Answer: Quotient is x 2 + x + 1. Remainder is 5. To check this answer, we must show that DQ + R equals the dividend. In other words, Does (x + 2)(x 2 + x + 1) + 5 = x 3 + 3x 2 + 3x + 7? All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

32 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Subtract x 3 + 2x 2 from the dividend to get x 2 + 3x + 7, which we call Diff. If degree of Diff is less than degree of the divisor, Diff is the remainder: Stop. If not, repeat the above process using Diff as the new dividend. Click slowly through the following demonstration. x 2 + x +1 x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 x 2 + 3x + 7 x 2 + 2x x + 7 x x into x 3 is x 2. Multiply x 2 by x + 2 Subtract to get Diff = x 2 + 3x + 7. x into x 2 is x. Multiply x by x + 2 Subtract to get Diff = x + 7. x into x is 1. Multiply 1 by x + 2 Subtract to get Diff = 5. At this point we stop because the degree 0 of 5 is less than the degree 1 of the divisor x + 2. Answer: Quotient is x 2 + x + 1. Remainder is 5. To check this answer, we must show that DQ + R equals the dividend. In other words, Does (x + 2)(x 2 + x + 1) + 5 = x 3 + 3x 2 + 3x + 7? Expand and rewrite the left side: (x + 2)(x 2 + x + 1) + 5 = x(x 2 + x + 1) + 2(x 2 + x + 1) + 5 = x 3 + x 2 + x + 2x 2 + 2x = x 3 + 3x 2 + 3x + 7 = the Dividend, as desired. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

33 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Subtract x 3 + 2x 2 from the dividend to get x 2 + 3x + 7, which we call Diff. If degree of Diff is less than degree of the divisor, Diff is the remainder: Stop. If not, repeat the above process using Diff as the new dividend. Click slowly through the following demonstration. x 2 + x +1 x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 x 2 + 3x + 7 x 2 + 2x x + 7 x x into x 3 is x 2. Multiply x 2 by x + 2 Subtract to get Diff = x 2 + 3x + 7. x into x 2 is x. Multiply x by x + 2 Subtract to get Diff = x + 7. x into x is 1. Multiply 1 by x + 2 Subtract to get Diff = 5. At this point we stop because the degree 0 of 5 is less than the degree 1 of the divisor x + 2. Answer: Quotient is x 2 + x + 1. Remainder is 5. To check this answer, we must show that DQ + R equals the dividend. In other words, Does (x + 2)(x 2 + x + 1) + 5 = x 3 + 3x 2 + 3x + 7? Expand and rewrite the left side: (x + 2)(x 2 + x + 1) + 5 = x(x 2 + x + 1) + 2(x 2 + x + 1) + 5 = x 3 + x 2 + x + 2x 2 + 2x = x 3 + 3x 2 + 3x + 7 = the Dividend, as desired. We can phrase the problem in a different way. A fraction of polynomials is called proper if the degree of the numerator is less than the degree of the denominator. Example 14: Rewrite x3 + 3x 2 + 3x + 7 as a x + 2 polynomial plus a proper fraction of polynomials. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

34 1.1.7 The long division algorithm Example 13: Use long division with remainder to divide x 3 + 3x 2 + 3x + 7 by x + 2. Solution: The dividend is x 3 + 3x 2 + 3x + 7. The Divisor is x + 2. Divide the Divisor s leading term into the dividend s leading term: x into x 3 is x 3 1 = x 2, which is written in the Quotient. Multiply x 2 by the Divisor to get x 2 (x + 2) = x 3 + 2x 2. Subtract x 3 + 2x 2 from the dividend to get x 2 + 3x + 7, which we call Diff. If degree of Diff is less than degree of the divisor, Diff is the remainder: Stop. If not, repeat the above process using Diff as the new dividend. Click slowly through the following demonstration. x 2 + x +1 x + 2 / x 3 + 3x 2 + 3x + 7 x 3 + 2x 2 x 2 + 3x + 7 x 2 + 2x x + 7 x x into x 3 is x 2. Multiply x 2 by x + 2 Subtract to get Diff = x 2 + 3x + 7. x into x 2 is x. Multiply x by x + 2 Subtract to get Diff = x + 7. x into x is 1. Multiply 1 by x + 2 Subtract to get Diff = 5. At this point we stop because the degree 0 of 5 is less than the degree 1 of the divisor x + 2. Answer: Quotient is x 2 + x + 1. Remainder is 5. To check this answer, we must show that DQ + R equals the dividend. In other words, Does (x + 2)(x 2 + x + 1) + 5 = x 3 + 3x 2 + 3x + 7? Expand and rewrite the left side: (x + 2)(x 2 + x + 1) + 5 = x(x 2 + x + 1) + 2(x 2 + x + 1) + 5 = x 3 + x 2 + x + 2x 2 + 2x = x 3 + 3x 2 + 3x + 7 = the Dividend, as desired. We can phrase the problem in a different way. A fraction of polynomials is called proper if the degree of the numerator is less than the degree of the denominator. Example 14: Rewrite x3 + 3x 2 + 3x + 7 as a x + 2 polynomial plus a proper fraction of polynomials. Solution: Do the long division. Then the problem asks for Dividend = Quotient + Remainder Divisor Divisor = x 2 + x x + 2. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 21

35 Factoring numbers and polynomials The word factor has two related meanings: Used as a verb, it means: rewrite as a product. Used as a noun, it means: one of the things being multiplied to form the product. 20 is the product of 5 and is a factorization of 20 with factors 5 and 4. x 3 + 4x 2 + 4x is the product of x and x 2 + 4x + 4 x(x 2 + 4x + 4) and (x + 2)x(x + 2) are factorizations of x 3 + 4x 2 + 4x. To factor a whole number, write it as a product of whole numbers. 7 = 3 7 is not a factorization of the 3 whole number 7. To factor a polynomial with integer coefficients, write it as a product of polynomials with integer coefficients. x 2 3 = (x + 3)(x 3) is not (in this course) a factorization of x 2 3. Definition of prime numbers and polynomials A number or polynomial is prime if it can t be factored. composite if it can be factored. A prime power factorization of a polynomial is one that is written as a product of of powers of distinct primes. The definition of prime requires clarification. Although 2 = 2 1 and x = x 1, these trivial factorizations don t count: x and 2 are both prime. For technical reasons that will not be explained here, 1 is not prime. Example 15: 2, 3, 5, 7, 11, 13, 17, 19 are the first 8 prime numbers. x, x + 2, 2x 3,, x 2 + x + 1, y, y 2 + 1, x + y, and xy + z are prime polynomials. Example 16: is a prime power factorization of 20. x(x + 2) 2 is a prime power factorization of x 3 + 4x 2 + 4x. Factoring has two main uses: To reduce a fraction, factor numerator and denominator and cancel common factors. To solve a polynomial equation, first move everything to one side and factor that side. Until further notice, number means positive integer and the coefficient of a polynomial s highest degree term is assumed to be positive. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 22

36 1.1.9 Factoring polynomials in one letter To factor a polynomial in x, we need to know How to factor x n from x m Write x m = x n x m n Example 17: Factor out x 6 from x 8 : Solution: x 8 = x 6 x 8 6 = x 6 x 2. Reminder: In the monomial 3xy 2, the coefficient is 3 and the letter product is xy 2. Definitions of Greatest Common Factor (GCF) The GCF of numbers is the largest number that is a factor of all of them. The GCF of letter products is the highest degree letter product that is a factor of all of them. How to factor out the GCF of a polynomial in x Factor out the GCF of the coefficients. Factor out the lowest power of x. Omit this step if the polynomial includes a constant term, which is a monomial that is just a number. Example 18: Factor out the GCF of each polynomial. First, polynomials with a constant term: 2x + 4 = 2(x + 2) 3x 2 + 6x + 12 = 3(x 2 + 2x + 4) 60x x = 10(6x 2 + 4x + 100) = 10 2(3x 2 + 2x + 50) = 20(3x 2 + 2x + 50) You could have done this in one step: 60x x = 20(3x 2 + 2x + 50). Stop here: no number (other than 1) is a common factor of 3, 2, and 50. Therefore 20 is the GCF of 60, 40, Here s how to factor polynomials with no constant term: 2x 8 + 4x 6 + 6x 7 = 2(x 8 + 2x 6 + 3x 7 ): Lowest x-power is x 6 = 2x 6 (x x x 7 6 ) Factor out x 6 = 2x 6 (x 2 + 2x 0 + 3x 1 ) = 2x 6 (x 2 + 3x + 2) 100x x 2 64x 5 GCF of coeffs is 4. = 4(25x 3 + 8x 2 16x 5 ) Lowest x-power is x 2 = 4x 2 (25x x 3 ) = 4x 2 ( 16x x + 8) or, if you factor out 1, 4x 2 (16x 3 25x 8) All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 23

37 Factoring quadratic polynomials Definitions for quadratic polynomials A quadratic polynomial is a polynomial ax 2 + bx + c with integer coefficients and a 0. The discriminant of ax 2 + bx + c is D = b 2 4ac A number is a perfect square if it is the square of some integer. The first few perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. How to factor a difference of squares For any expressions A and B A 2 B 2 = (A + B)(A B) Example 19: Factor each polynomial by rewriting it as a difference of squares Factor A B A 2 B 2 = (A + B)(A B) x x 3 x 2 9 = (x + 3)(x 3) (2x) x 3 4x 2 9 = (2x + 3)(2x 3) 3 2 x 2 3 x 9 x 2 = (3 + x)(3 x) 5 2 (2x) 2 5 2x 25 4x 2 = (5 + 2x)(5 2x) Most polynomials don t factor, so we need to know How to decide if a quadratic polynomial factors. Assume a, b, c have GCF 1. Then ax 2 + bx + c factors if D = b 2 4ac is a perfect square. does not factor if D is not a perfect square. Example 20: Decide if each polynomial factors: x 2 + 8x + 15 factors because D = = 4 = 2 2 is a perfect square x 2 + 4x + 2 does not factor because D = = 8 is not a perfect square. 6x x + 6 factors because D = = 25 = 5 2 is a perfect square. If you decide that ax 2 + bx + c factors, how do you find a factorization? The answer comes in two stages. How to factor ax 2 + bx + c when a = 1 Assume D = b 2 4ac is a perfect square. To factor x 2 + bx + c, find integers r, s with b = r + s and c = rs. Then x 2 + bx + c = (x + r)(x + s). General advice: if a < 0, factor out 1 from ax 2 + bx + c. For example, to factor x 2 + bx + c, rewrite it as 1(x 2 bx c) and factor x 2 bx c. All rights reserved. Copyright 2016 by Stanley Ocken CCNY Math Review Section 1.1: Fundamental operations 8/09/16 Frame 24