Factoring Polynomials

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Factoring Polynomials"

Transcription

1 Factoring Polynomials Any Any Any natural number that that that greater greater than than than 1 1can can 1 be can be be factored into into into a a a product of of of prime prime numbers. For For For example 0 0 = 0 = = ()()(5) and and30 and 30 = 30 = ()(3)(5) /\ /\ Lf..5. IS /\ /\ 35 In Inth th th chapter chapter we ll we ll we ll learn learn an an analogous way way way to to factor factor polynomials. Fundamental Theorem of of of Algebra AA Amonic monic polynomial a a a polynomial whose whose whose leading leading coe coecoefficient cient cient equals equals equals So1. So So x 4 x 4 x x 3 +5x 3 +5x 7monic,andx monic,but3x 4notmonic. x3 + 5x 7 x 3x 4 monic. The Carl Carl following Friedrich Friedrich result Gauss Gauss tells was was usthe how the boy to boy who factor who polynomials. dcovered a really a really It essentially quick quick way way tells to to see see usthat what that 1 + the prime =5050. polynomials + = are: In In1799, 1799, a a grown-up Gauss Gauss proved proved the the following following theorem: theorem: Any polynomial the product of a real number, Any and a Any polynomial collection polynomial the of monic the product quadratic product of polynomials of a real a real number, number, that and a collection of monic quadratic polynomials that do not and have a collection roots, and of monic of monic quadratic linear polynomials. that do do not not have have roots, roots, and and of of monic monic linear linear polynomials. polynomials. Th Th result called the Fundamental Theorem of Algebra. It one of the most Thresult important result called calledthe resultsthe Fundamental in Fundamental Theorem all of mathematics, Theoremof though of Algebra. Algebra. It from the It one form one of of the it s themost written most important inimportant results above, it sresults in probably inallall of di of mathematics, cult mathematics, though to immediately thoughfrom understand fromthetheform itsform it s importance. it swritten writtenin inabove, it s it sprobably probablydi difficult to toimmediately understand understanditsits importance. The importance. The explanation explanation for for why th th theorem theorem true true somewhat somewhat di di cult, and and it it beyond The beyond the explanation the scope scope of for of thwhy th course. th course. We ll theorem We ll have have to true to accept somewhat accept it it on on faith. difficult, and it beyond the scope of th course. We ll have to accept it on faith. faith. Examples. Examples. Examples. 4x 4x 1x 1x 4x +8canbefactoredintoaproductofanumber,4,and 1x a of a number, and two twomonic moniclinear linear polynomials, x x 11and andx x.. That That,, 4x 4x two 1x monic 1x linear polynomials, 1)(x 1)(x ). x 1 and x. That, 4~_ 1x+8=4(x 1)(x ). )

2 x 5 +x 4 7x 3 +14x 10x +0canbefactoredintoaproduct of a number, 1, a monic linear polynomial, x, and two monic quadratic polynomials that don t have roots, x + and x +5. That x 5 +x 4 7x 3 +14x 10x +0= (x )(x +)(x +5). (We can check the dcriminants of x +andx +5toseethat these two quadratics don t have roots.) x 4 x 3 +14x 6x +4=(x +3)(x x + 4). Again, x +3 and x x +4donothaveroots. Notice that in each of the above examples, the real number that appears in the product of polynomials 4 in the first example, 1inthesecond, and in the third the same as the leading coe cient for the original polynomial. Th always happens, so the Fundamental Theorem of Algebra can be more precely stated as follows: If p(x) =a n x n + a n 1 x n a 0, then p(x) theproductoftherealnumbera n, and a collection of monic quadratic polynomials that do not have roots, and of monic linear polynomials. Completely factored Apolynomialcompletely factored if it written as a product of a real number (which will be the same number as the leading coe cient of the polynomial), and a collection of monic quadratic polynomials that do not have roots, and of monic linear polynomials. Looking at the examples above, 4(x 1)(x ) and (x )(x +)(x +5) and (x +3)(x x +4)arecompletelyfactored. One reason it s nice to completely factor a polynomial because if you do, then it s easy to read o what the roots of the polynomial are. Example. Suppose p(x) = x 5 +10x 4 +x 3 38x +4x 48. Written in th form, its di cult to see what the roots of p(x) are. Butafterbeing completely factored, p(x) = (x +)(x 3)(x 4)(x +1). Therootsof 158

3 th polynomial can be read from the monic linear factors. They are, 3, and 4. (Notice that p(x) = (x +)(x 3)(x 4)(x +1)completelyfactored because x +1hasnoroots.) * * * * * * * * * * * * * Factoring linears To completely factor a linear polynomial, just factor out its leading coe - cient: ax + b = a x + b a For example, to completely factor x +6,writeitastheproduct(x +3). Factoring quadratics What a completely factored quadratic polynomial looks like will depend on how many roots it has. 0Roots.If the quadratic polynomial ax + bx + c has 0 roots, then it can be completely factored by factoring out the leading coe cient: ax + bx + c = a x + b a x + c a (The graphs of ax +bx+c and x + b a x+ c a di er by a vertical stretch or shrink that depends on a. A vertical stretch or shrink of a graph won t change the number of x-intercepts, so x + b a x + c a won t have any roots since ax + bx + c doesn t have any roots. Thus, x + b a x + c a completely factored.) Example. The dcriminant of 4x x+ equals ( ) 4(4)() = 4 3 = 8, a negative number. Therefore, 4x x + has no roots, and it completely factored as 4(x 1 x + 1 ). Roots. If the quadratic polynomial ax + bx + c has roots, we can name them 1 and. Roots give linear factors, so we know that (x 1 ) 159

4 and (x )arefactorsofax + bx + c. That means that there some polynomial q(x) suchthat ax + bx + c = q(x)(x 1 )(x ) The degree of ax + bx + c equals. Because the sum of the degrees of the factors equals the degree of the product, we know that the degree of q(x) plus the degree of (x 1 )plusthedegreeof(x ) equals. In other words, the degree of q(x) plus1plus1equals. Zero the only number that you can add to to get, so q(x) must have degree 0, which means that q(x) justaconstantnumber. Because the leading term of ax + bx + c namely ax theproductof the leading terms of q(x), (x 1 ), and (x ) namelythenumberq(x), x, andx itmustbethatq(x) =a. Therefore, ax + bx + c = a(x 1 )(x ) Example. The dcriminant of x +4x equals4 4()( ) = = 3, a positive number, so there are two roots. We can use the quadratic formula to find the two roots, but before we do, it s best to simplify the square root of the dcriminant: p 3 = p (4)(4)() = 4 p. Now we use the quadratic formula to find that the roots are and 4+4 p () 4 4 p () = 4+4p 4 = 4 4p 4 = 1+ p = 1 Therefore, x +4x completelyfactoredas x ( 1+ p p ) x ( 1 ) =(x +1 p p )(x +1+ p ) 1Root.If ax + bx + c has exactly 1 root (let s call it 1 )then(x afactorofax + bx + c. Hence, ax + bx + c = g(x)(x 1 ) for some polynomial g(x) )

5 Because the degree of a product the sum of the degrees of the factors, g(x) mustbeadegree1polynomial,anditcanbecompletelyfactoredinto something of the form (x ) where, R. Therefore, ax + bx + c = (x )(x 1 ) Notice that a root of (x )(x 1 ), so a root of ax + bx + c since they are the same polynomial. But we know that ax + bx + c has only one root, namely 1, so must equal 1. That means that ax + bx + c = (x 1 )(x 1 ) The leading term of ax +bx+c ax. The leading term of (x 1 )(x 1 ) x. Since ax + bx + c equals (x 1 )(x 1 ), they must have the same leading term. Therefore, ax = x. Hence, a =. Replace with a in the equation above, and we are left with ax + bx + c = a(x 1 )(x 1 ) Example. The dcriminant of 3x 6x+3 equals ( 6) 4(3)(3) = = 0, so there exactly one root. We find the root using the quadratic formula: ( 6) + p 0 (3) = 6 6 =1 Therefore, 3x 6x +3completelyfactoredas3(x 1)(x 1). Summary. The following chart summarizes the dcussion above. roots of ax + bx + c completely factored form of ax + bx + c no roots a(x + b a x + c a ) roots: 1 and a(x 1 )(x ) 1root: 1 a(x 1 )(x 1 ) 161

6 * * * * * * * * * * * * * Factors in Z Recall that the factors of an integer n are all of the integers k such that n = mk for some third integer m. Examples. 1 = 3 4, so 4 a factor of = 15, so 15 a factor of 30. 1, 1, n and n are all factors of an integer n. That s because n = n 1andn =( n)( 1). Important special case. If 1,,... n Z, then each of these numbers are factors of the product 1 n. For example,, 10, and 7 are each factors of 10 7=140. Check factors of degree 0 coe roots If k, 1,and are all integers, then the polynomial cient when searching for q(x) =k(x 1 )(x )=kx k( 1 + )x + k 1 has 1 and as roots, and each of these roots are factors of the degree 0 coe cient of q(x). (The degree 0 coe cient k 1.) More generally, if k, 1,,..., n Z, then the degree 0 coe polynomial g(x) =k(x 1 )(x ) (x n ) cient of the equals k 1 n. That means that each of the roots of g(x) whichare the i arefactorsofthedegree0coe cient of g(x). Now it s not true that every polynomial has integer roots, but many of the polynomials you will come across do, so the two paragraphs above o er a powerful hint as to what the roots of a polynomial might be. 16

7 When searching for roots of a polynomial whose coe cients are all integers, check the factors of the degree 0 coe cient. Example. 3and 7arebothrootsof(x 3)(x +7). Notice that (x 3)(x +7)=x +8x 4, and that 3 and 7areboth factors of 4. Example. Suppose p(x) =3x 4 +3x 3 3x +3x 6. Th a degree 4 polynomial, so it will have at most 4 roots. There n t a really easy way to find the roots of a degree 4 polynomial, so to find the roots of p(x), we have to start by guessing. The degree 0 coe cient of p(x) 6, so a good place to check for roots in the factors of 6. The factors of 6are1, 1,,, 3, 3, 6, and 6, so we have eight quick candidates for what the roots of p(x) mightbe. Aquickcheckshows that of these eight candidates, exactly two are roots of p(x) namely 1 and. That to say, p(1) = 0 and p( ) = 0. * * * * * * * * * * * * * Factoring cubics It follows from the Fundamental Theorem of Algebra that a cubic polynomial either the product of a constant and three linear polynomials, or else it the product of a constant, one linear polynomial, and one quadratic polynomial that has no roots. In either case, any cubic polynomial guaranteed to have a linear factor, and thus guaranteed to have a root. You re going to have to guess what that root by looking at the factors of the degree 0 coe cient. (There a cubicformula thatlikethequadraticformulawilltellyoutherootsof acubic,buttheformuladi cult to remember, and you d need to know about complex numbers to be able to use it.) Once you ve found a root, factor out the linear factor that the root gives you. You will now be able to write the cubic as a product of a monic linear 163

8 factors of 3 are 1, 1, 3, and 3. Check these factors to see if any of them are roots. After checking, you ll see that 1 a root. That means that x 1 a factor of x3 3x + 4x 3. Therefore, we can divide x3 3x + 4x 3 by x 1 polynomial to get another and polynomial a quadratic polynomial. Completely factor the quadratic and then polynomial you will and have a quadratic completely x3 3x+4x 3 polynomial. factored the Completely cubic. factor the quadratic and then you will have completely factored the =x cubic. x+3 Problem. Completely factor x 3 3x +4x 3. Problem. Thus, Completely factor x3 3~ + 4x 3. Solution. Start x3 3x+4x 3=(x 1)(x x+3) by guessing a root. The degree 0 coe cient 3, and the factors Solution. of 3are1, Start by guessing 1, 3, anda root. 3. Check The these degreefactors 0 coefficient to see if any 3, of and them the are factors roots. of 3 are 1, 1, 3, and 3. Check these to see if any of them are After roots. checking, you ll see that 1 a root. That means that x 1afactor of After x 3 checking, 3x +4x you ll 3. Therefore, see that 1we acan root. divide That xmeans 3 3x that +4xx 13byx a factor 1 to of get x3 another 3x + polynomial 4x 3. Therefore, divide x3 3x + 4x 3 by x 1 to get another polynomial x 3 3x +4x 3 =x x +3 The dcriminant ofx3 3x+4x 3 x x x equals (_1) 4()(3) = 1 4 = =x x+3 3, Thus, a negative number. Therefore, x x + 3 has no roots, so to completely Thus, factor x x 3 3x +4x 3 = (x 1)(x x +3 we just have to factor out the leading x +3) coefficient as follows: x x-1-3=(x x3 3x+4x 3=(x 1)(x x+3) ~x+~). x3-3x1 ~~+z-3 (x-l) /\ a (zx_x+3 ) /N The dcriminant of x x +3equals( 1) 4()(3) = 1 4 = 3, anegativenumber. The dcriminant of Therefore,x x x + 3 equals x +3(_1) has no 4()(3) roots, so = to 1 completely 4 = 3, factor a negative x number. x +3 we just Therefore, have tox factor x out + 3the has leading no roots, coe so cient to as completely follows: x factor The x x final +3= answer x 1 x + 3. x +3 we just have to factor out leading coefficient as follows: x x-1-3=(x ~x+~). (x_1)(x_ ~+~) 137 x3-3x1 ~~+z-3 (x-l) /\ a (zx_x+3 ) /N The final answer The final answer (x 1) x 1 x + 3 (x_1)(x_ ~+~)

9 3x3 3x 15x-I-6 = (x+)(3x 9x+3) Problem. Completely factor 3x 3 3x 15x +6. Problem. Completely factor 3w3 3w 15w + 6. Solution. The factors of 6 are {1, 1,,, 3, 3, 6, 6}. Check to see that Solution. aroot.thendividebyx The factors of 6 are {1, +tofindthat 1,,,3, 3,6, 6}. Check to see that The dcriminant of 3w 9w +3 equals 45, and thus 3w 9w +3 has two a root. Then divide by x + to find that roots and can be factored 3x 3 further. 3x 15x +6 The leading coefficient 3w3 of = 3x 3x 9x +3 x 3w 15w + 9w , and we can use the quadratic = 3w 9x+3 formula so to check that the roots x+of 3w 9w + 3 are ~ and~ From what we so learned above 3x about 3 3x factoring 15x +6 quadratics, = (x +)(3x we know 9x that +3) 3w 9w + 3 3(w ~~ ~)(x ~ 3x3 3x 15x-I-6 V s) = (x+)(3x 9x+3) 3x3-3x-15i /\ (x+~ (3x_cbc. / The dcriminant of 3x 39x +3equals45, c~ / 3+S\,~i apositivenumber, / andthus 3x The dcriminant 9x +3hastworootsandcanbefactoredfurther. of 3w 9w +3 equals 45, and thus 3w 9w +3 has two roots The and leading can be coe factored cient further. of 3x 9x +33,andwecanusethequadratic formula To The summarize, leading coefficient of 3w to check that the roots of 3x 9x+3 are 3+p 5 and 3 p 9w + 3 3, and we can use the 5 quadratic. From what formula to check that the roots of 3w 9w + 3 are ~ and~ we learned 3w3 above about factoring quadratics, we know that 3x From what 9x +3= 3+ 3(x p 5 3 )(x p 3w 15w + 6 = (w + )(3w we learned above about 9w + 3) 5 factoring quadratics, we know that 3w 9w + 3 ). 3(w ~~ ~)(x ~ V s) I 3+~/~~i 3 Vg =(w-i-)3~çw )çw 3x3-3x-15i / 3+v g ~,i 3 ~./~ =3(w+)~w )k~x (x+~ /\ (3x_cbc. / / 3+S\ / 3 c~,~i To summarize, To summarize, 3x 3 3x 15x +6=(x +)(3x 9x +3) 3w3 3+ p 3w 15w + 6 = (w + )(3w 9w + 3) 5 p 3 5 =(x +)3I x 3+~/~~i x 3 Vg =(w-i-)3~çw 3+ p )çw 5 p 3 5 =3(x +) / x 3+v g ~,i x 3 ~./~ =3(w+)~w )k~x 165

10 Factoring quartics Degree 4 polynomials are tricky. As with cubic polynomials, you should begin by checking whether the factors of the degree 0 coe cient are roots. If one of them a root, then you can use the same basic steps that we used with cubic polynomials to completely factor the polynomial. The problem with degree 4 polynomials that there s no reason that a degree 4 polynomial has to have any roots take (x +1)(x +1) for example. Because a degree 4 polynomial might not have any roots, it might not have any linear factors, and it s very hard to guess which quadratic polynomials it might have as factors. * * * * * * * * * * * * * 166

11 Exerces Completely factor the polynomials given in #1-8 1.) 10x +0.) x +5 3.) x 1x 18 4.) 10x +3 5.) 3x 10x +5 6.) 3x 4x +5 7.) x +6x 3 8.) 5x +3x 9.) Find a root of x 3 5x +10x ) Find a root of 15x 3 +35x +30x ) Find a root of x 3 x x 3. Completely factor the polynomials in # ) x 3 x + x ) 5x 3 9x +8x 0 14.) x 3 +17x 3 15.) 4x 3 0x +5x 3 16.) x 4 5x ) How can the Fundamental Theorem of Algebra be used to show that any polynomial of odd degree has at least one root? 167

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations

More information

3 1. Note that all cubes solve it; therefore, there are no more

3 1. Note that all cubes solve it; therefore, there are no more Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

More information

Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

3.2 The Factor Theorem and The Remainder Theorem

3.2 The Factor Theorem and The Remainder Theorem 3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

UNCORRECTED PAGE PROOFS

UNCORRECTED PAGE PROOFS number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.

More information

1 Shapes of Cubic Functions

1 Shapes of Cubic Functions MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic

More information

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11. 9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

More information

3.6 The Real Zeros of a Polynomial Function

3.6 The Real Zeros of a Polynomial Function SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

More information

College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran

College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran College Algebra - MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or x-intercept) of a polynomial is identical to the process of factoring a polynomial.

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

2.4 Real Zeros of Polynomial Functions

2.4 Real Zeros of Polynomial Functions SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower

More information

Unit 3: Day 2: Factoring Polynomial Expressions

Unit 3: Day 2: Factoring Polynomial Expressions Unit 3: Day : Factoring Polynomial Expressions Minds On: 0 Action: 45 Consolidate:10 Total =75 min Learning Goals: Extend knowledge of factoring to factor cubic and quartic expressions that can be factored

More information

Solving Cubic Polynomials

Solving Cubic Polynomials Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial

More information

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005 Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials 4-1-2014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall

More information

FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

Section 3.2 Polynomial Functions and Their Graphs

Section 3.2 Polynomial Functions and Their Graphs Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)

More information

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

More information

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date Leaving Certificate Honours Maths - Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many

More information

Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Factoring 4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

REVIEW EXERCISES DAVID J LOWRY

REVIEW EXERCISES DAVID J LOWRY REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions 1 2.1. Factoring and Solving Quadratics 1 2.2. Polynomial Inequalities 3 2.3. Rational Functions 4 2.4. Exponentials and

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

More information

Factoring Cubic Polynomials

Factoring Cubic Polynomials Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

More information

minimal polyonomial Example

minimal polyonomial Example Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

More information

calculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0,

calculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0, Homework #02, due 1/27/10 = 9.4.1, 9.4.2, 9.4.5, 9.4.6, 9.4.7. Additional problems recommended for study: (9.4.3), 9.4.4, 9.4.9, 9.4.11, 9.4.13, (9.4.14), 9.4.17 9.4.1 Determine whether the following polynomials

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

More information

3.6. The factor theorem

3.6. The factor theorem 3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the

More information

Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

Academic Success Centre

Academic Success Centre 250) 960-6367 Factoring Polynomials Sometimes when we try to solve or simplify an equation or expression involving polynomials the way that it looks can hinder our progress in finding a solution. Factorization

More information

Factoring and Applications

Factoring and Applications Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

More information

Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial

More information

Polynomial Expressions and Equations

Polynomial Expressions and Equations Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

More information

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

More information

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x). .7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

More information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of. Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

More information

A. Factoring out the Greatest Common Factor.

A. Factoring out the Greatest Common Factor. DETAILED SOLUTIONS AND CONCEPTS - FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

More information

7.2 Quadratic Equations

7.2 Quadratic Equations 476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.

63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15. 9.4 (9-27) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27-in. wheel, 44 teeth

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of

More information

Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

Discrete Mathematics: Homework 7 solution. Due: 2011.6.03 EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project

FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project 9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module

More information

Section 6.1 Factoring Expressions

Section 6.1 Factoring Expressions Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

More information

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

More information

0.4 FACTORING POLYNOMIALS

0.4 FACTORING POLYNOMIALS 36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use

More information

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

8 Polynomials Worksheet

8 Polynomials Worksheet 8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

More information

H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

More information

Roots of Polynomials

Roots of Polynomials Roots of Polynomials (Com S 477/577 Notes) Yan-Bin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x

More information

15. Symmetric polynomials

15. Symmetric polynomials 15. Symmetric polynomials 15.1 The theorem 15.2 First examples 15.3 A variant: discriminants 1. The theorem Let S n be the group of permutations of {1,, n}, also called the symmetric group on n things.

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

More information

The degree of a polynomial function is equal to the highest exponent found on the independent variables.

The degree of a polynomial function is equal to the highest exponent found on the independent variables. DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

A Systematic Approach to Factoring

A Systematic Approach to Factoring A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

More information

Factoring Polynomials

Factoring Polynomials Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,

More information

SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014))

SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) There are so far 8 most common methods to solve quadratic equations in standard form ax² + bx + c = 0.

More information

The Factor Theorem and a corollary of the Fundamental Theorem of Algebra

The Factor Theorem and a corollary of the Fundamental Theorem of Algebra Math 421 Fall 2010 The Factor Theorem and a corollary of the Fundamental Theorem of Algebra 27 August 2010 Copyright 2006 2010 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Aside

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

More information

Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions Procedure for Graphing Polynomial Functions P(x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

More information

7. Some irreducible polynomials

7. Some irreducible polynomials 7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of

More information

Algebra II A Final Exam

Algebra II A Final Exam Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

More information

is the degree of the polynomial and is the leading coefficient.

is the degree of the polynomial and is the leading coefficient. Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

Factoring Trinomials: The ac Method

Factoring Trinomials: The ac Method 6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

More information

MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

More information

Winter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com

Winter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).

More information