Algebra II Notes, Chapter Four, Products and Factors of Polynomials

ALGEBRA II BIBLE – Chapter Four

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ChapterChapterChapterChapter Four (Products and Factors of Polynomials Four (Products and Factors of Polynomials Four (Products and Factors of Polynomials Four (Products and Factors of Polynomials))))

Chapter Four, Section OneChapter Four, Section OneChapter Four, Section OneChapter Four, Section One (Polynomials) (Polynomials) (Polynomials) (Polynomials)

CONTENTS Working with Polynomials

4444----1111 Polynomials 4444----2222 Using Laws of Exponents 4444----3333 Multiplying Polynomials

Factors of Polynomials

4444----4444 Using Prime Factorization 4444----5555 Factoring Polynomials 4444----6666 Factoring Quadratic Polynomials

Applications of Factoring

4444----7777 Solving Polynomial Equations 4444----8888 Problem Solving Using Polynomial Equations 4444----9999 Solving Polynomial Inequalities

This particular set of notes is from Algebra and Trigonometry Structure and Method - Book 2. All credit is to be given to the authors and publishers of said book. The study guide made from the book contains definitions, diagrams, and notes taken directly from the book.

Constant:Constant:Constant:Constant: a number Monomial:Monomial:Monomial:Monomial: a constant, a variable, or a product of a constant and one or more variables Coefficient Coefficient Coefficient Coefficient ((((or numerical coefficient):numerical coefficient):numerical coefficient):numerical coefficient): the constant (or numerical) factor in a monomial Degree of a variable in a monomial:Degree of a variable in a monomial:Degree of a variable in a monomial:Degree of a variable in a monomial: the number of times the variable occurs as a factor in the monomial Degree of a monomial:Degree of a monomial:Degree of a monomial:Degree of a monomial: the sum of the degrees of the variables in the monomial. A nonzero constant has degree 0. The constant 0 has no degree.

Similar (Similar (Similar (Similar (or like) monomials: like) monomials: like) monomials: like) monomials: monomials that are identical or that differ only in their coefficients Polynomial:Polynomial:Polynomial:Polynomial: a monomial or a sum of monomials. The monomials in a polynomial are called the termstermstermsterms of the polynomial. Simplified polynomial:Simplified polynomial:Simplified polynomial:Simplified polynomial: a polynomial in which no two terms are similar. The terms are usually arranged in order of decreasing degree of one of the variables. Degree of a polynomial:Degree of a polynomial:Degree of a polynomial:Degree of a polynomial: the greatest of the degrees of its terms after it has been simplified.

Adding and Subtracting PolynomialsAdding and Subtracting PolynomialsAdding and Subtracting PolynomialsAdding and Subtracting Polynomials To add two or more polynomials, write their sum and then simplify by combining similar terms. To subtract one polynomial from another, add the opposite of each term of the polynomial you’re subtracting.


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Chapter Four, Section TwoChapter Four, Section TwoChapter Four, Section TwoChapter Four, Section Two (Using Laws of Exponents) (Using Laws of Exponents) (Using Laws of Exponents) (Using Laws of Exponents) Chapter Four, SectioChapter Four, SectioChapter Four, SectioChapter Four, Section Threen Threen Threen Three (Multiplying Polynomials) (Multiplying Polynomials) (Multiplying Polynomials) (Multiplying Polynomials) A binomialbinomialbinomialbinomial is a polynomial that has two terms. A trinomialtrinomialtrinomialtrinomial is a polynomial that has three terms. When multiplying two binomials, the FOIL method is used. FOIL reminds you to multiply the First, Outer, Inner, and Last terms when multiplying two binomials. Chapter Four, Section FourChapter Four, Section FourChapter Four, Section FourChapter Four, Section Four (Using Prime Factorization) (Using Prime Factorization) (Using Prime Factorization) (Using Prime Factorization) To factorfactorfactorfactor a number over a set of numbers, you write it as a product of numbers chosen from that set, called the factor setfactor setfactor setfactor set. A prime numberprime numberprime numberprime number, or primeprimeprimeprime, is an integer greater than 1 whose only positive integral factors are itself and 1. To find the prime factorizationprime factorizationprime factorizationprime factorization of a positive integer, you write the integer as a product of primes. The greatest common factor (GCF)greatest common factor (GCF)greatest common factor (GCF)greatest common factor (GCF) of two or more integers is the greatest integer that is a factor of each. The lllleast common multiple (LCM)east common multiple (LCM)east common multiple (LCM)east common multiple (LCM) of two or more integers is the least positive integer having each as a factor. The greatest common factor (GCF)greatest common factor (GCF)greatest common factor (GCF)greatest common factor (GCF) of two or more monomials is the common factor that has the greatest degree and the greatest numerical coefficient. The least least least least common multiple (LCM)common multiple (LCM)common multiple (LCM)common multiple (LCM) of two or more monomials is the common multiple that has the least degree and the least positive numerical coefficient.

Laws of ExponentsLaws of ExponentsLaws of ExponentsLaws of Exponents Let a and b be real numbers and m and n be positive integers. Then: 1. a1. a1. a1. ammmm •••• a a a annnn = a = a = a = am+nm+nm+nm+n 2. (ab)2. (ab)2. (ab)2. (ab) m m m m = a = a = a = ammmmbbbbmmmm 3. (a3. (a3. (a3. (ammmm)))) m m m m = a = a = a = ammmmnnnn

Special ProductSpecial ProductSpecial ProductSpecial Product Pattern and ExamplePattern and ExamplePattern and ExamplePattern and Example

(a + b)(a + b)(a + b)(a + b)2222 = a = a = a = a2222 + 2ab + b + 2ab + b + 2ab + b + 2ab + b2222 (first + second)2 = (first)2 + 2(first)(second) + (second)2

(4s + 3t)2 = (4s)2 + 2(4s)( 3t) + (3t)2

= 16s2 + 24st + 9t2

(a (a (a (a ---- b) b) b) b)2222 = a = a = a = a2222 ---- 2ab + b 2ab + b 2ab + b 2ab + b2222 (first - second)2 = (first)2 - 2(first)(second) + (second)2

(3x - 5)2 = (3x)2 - 2(3x)( 5) + 52

= 9x2 - 30t + 25

(a + b)(a (a + b)(a (a + b)(a (a + b)(a –––– b) = a b) = a b) = a b) = a2222 ---- b b b b2222 (first + second)(first – second) = (first)2 - (second)2

(2p – 3q) = (2p)2 – (3q)2

= 4p2 – 9q2


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Chapter Four, Section FiveChapter Four, Section FiveChapter Four, Section FiveChapter Four, Section Five (Factoring Polynomials) (Factoring Polynomials) (Factoring Polynomials) (Factoring Polynomials) To factorfactorfactorfactor a polynomial you express it as a product of other polynomials taken from a specified factor set. The first step in factoring a polynomial is to find its greatest greatest greatest greatest monomial factormonomial factormonomial factormonomial factor, that is, the GCF of its terms. The polynomials a2 + 2ab + b2 and a2 - 2ab + b2, which are the results of squaring a + b and a – b, respectively, are called perfect square trinomialsperfect square trinomialsperfect square trinomialsperfect square trinomials. Also, the polynomial a2 - b2, which is the product of a + b and a – b, is called a difference of squaresdifference of squaresdifference of squaresdifference of squares.... Chapter Four, Section SixChapter Four, Section SixChapter Four, Section SixChapter Four, Section Six (Factoring Quadratic Polynomials) (Factoring Quadratic Polynomials) (Factoring Quadratic Polynomials) (Factoring Quadratic Polynomials)

Polynomials of the form ax2 + bx + c (a ≠ 0) are called quadraticquadraticquadraticquadratic or secondsecondsecondsecond----degreedegreedegreedegree polynomialspolynomialspolynomialspolynomials. The term ax2 is the quadratic termquadratic termquadratic termquadratic term, bx is the linear termlinear termlinear termlinear term, and c is the constant termconstant termconstant termconstant term. A quadratic trinomialquadratic trinomialquadratic trinomialquadratic trinomial is a quadratic polynomial for which a, b, and c are all nonzero integers. If the quadratic trinomial ax2 + bx + c can be factored into the product (px +q)(rx + s) where p, q, r, and s are integers, then

ax2 + bx + c = (px + q)(rx + s) = prx2 + (ps + qr)x + qs Setting corresponding coefficients equal gives

a = pr, b = ps + qr, and c = qs

A polynomial that has more than one term and cannot be expressed as a product of polynomials of lower degree taken from a given factor set is said to be irreduirreduirreduirreduciblecibleciblecible over that set. An irreducible polynomial with integral coefficients is primeprimeprimeprime if the greatest common factor of its coefficients is one. A polynomial is factored completelyfactored completelyfactored completelyfactored completely when it is written as a product of factors and each factor is either a monomial, a prime polynomial, or a power of a prime polynomial.

Perfect Square TrinPerfect Square TrinPerfect Square TrinPerfect Square Trinomialsomialsomialsomials aaaa2222 + 2ab + b + 2ab + b + 2ab + b + 2ab + b2222 = (a + b)= (a + b)= (a + b)= (a + b)2222 aaaa2222 ---- 2ab + b 2ab + b 2ab + b 2ab + b2222 = (a = (a = (a = (a ---- b) b) b) b)2222

Difference of SquaresDifference of SquaresDifference of SquaresDifference of Squares aaaa2222 ---- b b b b2 2 2 2 = (a + b)(a = (a + b)(a = (a + b)(a = (a + b)(a –––– b) b) b) b)

Sums and Difference of CubesSums and Difference of CubesSums and Difference of CubesSums and Difference of Cubes aaaa3333 + b + b + b + b3 3 3 3 = (a + b)= (a + b)= (a + b)= (a + b)(a(a(a(a2222 –––– ab + b ab + b ab + b ab + b2222)))) aaaa3333 ---- b b b b3 3 3 3 = (a = (a = (a = (a ---- b)(a b)(a b)(a b)(a2222 + ab + b + ab + b + ab + b + ab + b2222))))


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The greatest common factor (GCF)greatest common factor (GCF)greatest common factor (GCF)greatest common factor (GCF) of two or more polynomials is the common factor having the greatest degree and the greatest constant factor. The least commonleast commonleast commonleast common multiple (LCM)multiple (LCM)multiple (LCM)multiple (LCM) of two or more polynomials is the common multiple having least degree and least positive constant factor. Chapter Four, Section SevenChapter Four, Section SevenChapter Four, Section SevenChapter Four, Section Seven (Solving Polynomial Equations) (Solving Polynomial Equations) (Solving Polynomial Equations) (Solving Polynomial Equations) A polynomial equationpolynomial equationpolynomial equationpolynomial equation is an equation is that is equivalent to one with a polynomial as one side and 0 as the other. A rootrootrootroot, or solutionsolutionsolutionsolution, of a polynomial equation is a value of the variable that satisfies the equation. The zero-product property is used to solve polynomial equations. It is also used to find zeros of polynomial functions. A number r is a zezezezerorororo of a function if f(r) = 0. If f(x) = x(x-6)2, x -6 occurs as a factor of f twice, which makes it a double zerodouble zerodouble zerodouble zero of the function f ad a double rootdouble rootdouble rootdouble root of the equation f(x) = 0. In general, the zeros and roots arising from repeated factors are called multipmultipmultipmultiple zerole zerole zerole zeros s s s of the functions and multiple multiple multiple multiple rootsrootsrootsroots of equations. Chapter Four, Section EightChapter Four, Section EightChapter Four, Section EightChapter Four, Section Eight (Problem Solving Using Polynomial Equations) (Problem Solving Using Polynomial Equations) (Problem Solving Using Polynomial Equations) (Problem Solving Using Polynomial Equations) An equation that represents a real life problem is called a mathematical modelmathematical modelmathematical modelmathematical model. Chapter Four, Section NineChapter Four, Section NineChapter Four, Section NineChapter Four, Section Nine (Solving Polyno (Solving Polyno (Solving Polyno (Solving Polynomial Inequalities)mial Inequalities)mial Inequalities)mial Inequalities) A polynomialpolynomialpolynomialpolynomial inequalityinequalityinequalityinequality is an inequality that is equivalent to an inequality with a polynomial on one side and 0 as the other side. You can often solve a polynomial inequality that has 0 as one side by factoring the polynomial into linear factors and applying one of the following facts:

ab > 0 if and only if a and b have the same signs ab < 0 if and only if a and b have opposite signs

A sign graph can be used to help find and graph the solution set of a polynomial inequality.

To use the zero-product property to solve a polynomial equation, you need to 1. write the equation with 0 as one side, 2. factor the other side of the equation, and 3. solve the equation obtained by setting each factor equal to 0.


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Chapter SummaryChapter SummaryChapter SummaryChapter Summary

1.1.1.1. A monomial is a constant, a variable, or a product of a constant and one or more variables. A polynomial is a monomial or a sum of monomials. A simplified polynomial has no two terms similar. The terms of a simplified polynomial are usually arranged in decreasing degree of one of the variables.

2.2.2.2. If a and b are real numbers and m and n are positive integers, then:

Law 1 am • an = am + n Law 2 (ab)m = am bm

Law 3 (a m)n = amn

3.3.3.3. To find the product of two polynomials, multiply each term of one polynomial by each term of the other and simplify the result.

4.4.4.4. A prime is an integer greater than 1 whose only positive integral factors are

itself and 1. To find the prime factorization of an integer, write it as the product of primes.

5.5.5.5. Use prime factorization to determine the greatest common factor (GCF) or

least common multiple (LCM) of two or more integers. To find the GCF, take the least power of each common prime factor in the factorization. To find the LCM, take the greatest power of each prime factor in the factorizations.

6.6.6.6. The GCF of two or more monomials is the factor of each that has the greatest

degree and greatest coefficient. The greatest monomial factor of a polynomial is the GCF of its terms.

7.7.7.7. The LCM of two or more monomials is the multiple of each that has the least

degree and least positive coefficient.

8.8.8.8. The following strategies are useful in factoring polynomials: Factor out the greatest monomial factor.

Look for special products such as a perfect square trinomial, a difference of squares, and a sum or difference of cubes.

Rearrange and group terms.

Factor quadratic trinomials into products of linear factors. 9.9.9.9. Many polynomial equations can be solved by factoring and using the zero-

product property. Many polynomial inequalities can be solved by factoring and determining the signs of the factors. If the polynomial has three or more factors, it may be helpful to use a sign graph to determine the solution.

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Algebra II Notes, Chapter Four, Products and Factors of Polynomials