7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of terms, with only nonnegative integer eponents permitted on the variables. If the terms of a polynomial contain only the variable, then the polynomial is called a polynomial in. (Polynomials in other variables are defined similarly.) Eamples of polynomials include 5 3 8 2 7 4, 9p 5 3, 8r 2, and 6. The epression 9 2 4 6 is not a polynomial because of the presence of 6. The terms of a polynomial cannot have variables in a denominator. The greatest eponent in a polynomial in one variable is the degree of the polynomial. A nonzero constant is said to have degree 0. (The polynomial 0 has no degree.) For eample, 3 6 5 2 2 3 is a polynomial of degree 6. A polynomial can have more than one variable. A term containing more than one variable has degree equal to the sum of all the eponents appearing on the variables in the term. For eample, 3 4 y 3 z 5 is of degree 4 3 5 12. The degree of a polynomial in more than one variable is equal to the greatest degree of any term appearing in the polynomial. By this definition, the polynomial is of degree 8 because of the 6 y 2 term. 2 4 y 3 3 5 y 6 y 2
7.6 Polynomials and Factoring 373 A polynomial containing eactly three terms is called a trinomial and one containing eactly two terms is a binomial. For eample, 7 9 8 4 1 is a trinomial of degree 9. The table shows several polynomials and gives the degree and type of each. Polynomial Degree Type 9p 7 4p 3 8p 2 29 11 8 15 10r 6 s 8 5a 3 b 7 3a 5 b 5 4a 2 b 9 a 10 7 Trinomial 15 Binomial 14 Monomial 11 None of these Addition and Subtraction Since the variables used in polynomials represent real numbers, a polynomial represents a real number. This means that all the properties of the real numbers mentioned in this book hold for polynomials. In particular, the distributive property holds, so 3m 5 7m 5 3 7m 5 4m 5. Like terms are terms that have the eact same variable factors. Thus, polynomials are added by adding coefficients of like terms; polynomials are subtracted by subtracting coefficients of like terms. EXAMPLE 1 Add or subtract, as indicated. 2y 4 3y 2 y 4y 4 7y 2 6y 2 4y 4 3 7y 2 1 6y 6y 4 4y 2 7y 3m 3 8m 2 4 m 3 7m 2 3 3 1m 3 8 7m 2 4 3 4m 3 15m 2 7 (c) 8m 4 p 5 9m 3 p 5 11m 4 p 5 15m 3 p 5 19m 4 p 5 6m 3 p 5 (d) 4 2 3 7 52 2 8 4 4 2 43 47 52 2 58 54 4 2 12 28 10 2 40 20 6 2 28 48 Distributive property Associative property Add like terms. As shown in parts,, and (d) of Eample 1, polynomials in one variable are often written with their terms in descending powers; so the term of greatest degree is first, the one with the net greatest degree is second, and so on. Multiplication The associative and distributive properties, together with the properties of eponents, can also be used to find the product of two polynomials. For eample, to find the product of 3 4 and 2 2 3 5, treat 3 4 as a single epression and use the distributive property as follows. 3 42 2 3 5 3 42 2 3 43 3 45
374 CHAPTER 7 The Basic Concepts of Algebra Now use the distributive property three separate times on the right of the equality symbol to get 3 42 2 3 5 32 2 42 2 33 43 35 45 6 3 8 2 9 2 12 15 20 6 3 17 2 27 20. It is sometimes more convenient to write such a product vertically, as follows. 2 2 3 5 3 4 8 2 12 20 6 3 9 2 15 6 3 17 2 27 20 k 42 2 3 5 k 32 2 3 5 Add in columns. EXAMPLE 2 Multiply 3p 2 4p 1p 3 2p 8. Multiply each term of the second polynomial by each term of the first and add these products. It is most efficient to work vertically with polynomials of more than two terms, so that like terms can be placed in columns. 3p 2 4p 1 p 3 2p 8 24p 2 32p 8 Multiply 3p 2 4p 1 by 8. 6p 3 8p 2 2p Multiply 3p 2 4p 1 by 2p. 3p 5 4p 4 p 3 Multiply 3p 2 4p 1 by p 3. 3p 5 4p 4 7p 3 32p 2 34p 8 Add in columns. The FOIL method is a convenient way to find the product of two binomials. The memory aid FOIL (for First, Outside, Inside, Last) gives the pairs of terms to be multiplied to get the product, as shown in the net eamples. The special product y y 2 y 2 can be used to solve some multiplication problems. For eample, 51 49 50 150 1 50 2 1 2 2500 1 2499 102 98 100 2100 2 100 2 2 2 10,000 4 9996. Once these patterns are recognized, multiplications of this type can be done mentally. EXAMPLE 3 Find each product. F O I L 6m 14m 3 6m4m 6m3 14m 13 24m 2 14m 3 2 72 7 4 2 14 14 49 4 2 49 In part of Eample 3, the product of two binomials was a trinomial, while in part the product of two binomials was a binomial. The product of two binomials of the forms y and y is always a binomial. Check by multiplying that the following is true. Product of the Sum and Difference of Two Terms y y 2 y 2
7.6 Polynomials and Factoring 375 This product is called the difference of two squares. Since products of this type occur frequently, it is important to be able to recognize when this pattern should be used. EXAMPLE 4 Find each product. 3p 113p 11 Using the pattern discussed above, replace with 3p and y with 11. (c) 3p 113p 11 3p 2 11 2 9p 2 121 5m 3 35m 3 3 5m 3 2 3 2 25m 6 9 9k 11r 3 9k 11r 3 9k 2 11r 3 2 81k 2 121r 6 The squares of binomials are also special products. Squares of Binomials y 2 2 2y y 2 y 2 2 2y y 2 y Area: 2 Area: y The special product y 2 2 2y y 2 can be illustrated geometrically using the diagram shown here. Each side of the large square has length y, so the area of the square is y 2. The large square is made up of two smaller squares and two congruent rectangles. The sum of the areas of these figures is 2 2y y 2. Area: y Area: y 2 Since these epressions represent the same quantity, they must be equal, thus giving us the pattern for squaring a binomial. y EXAMPLE 5 Find each product. 2m 5 2 2m 2 22m5 5 2 4m 2 20m 25 3 7y 4 2 3 2 237y 4 7y 4 2 9 2 42y 4 49y 8 As shown in Eample 5, the square of a binomial has three terms. Students often mistakenly give 2 y 2 as equivalent to the product y 2. Be careful to avoid that error. The process of finding polynomials whose product equals a given polynomial is called factoring. For eample, since 4 12 4 3, both 4 and 3 are called factors of 4 12. Also, 4 3 is called the factored form of 4 12. A polynomial that cannot be written as a product of two polynomials with integer coefficients is a prime polynomial. A polynomial is factored completely when it is written as a product of prime polynomials with integer coefficients. Factoring Out the Greatest Common Factor Some polynomials are factored by using the distributive property. For eample, to factor 6 2 y 3 9y 4 18y 5, we look for a monomial that is the greatest common factor of all the terms of the polynomial. For this polynomial, 3y 3 is the greatest common factor. By the distributive property, 6 2 y 3 9y 4 18y 5 3y 3 2 2 3y 3 3y 3y 3 6y 2 3y 3 2 2 3y 6y 2.
376 CHAPTER 7 The Basic Concepts of Algebra EXAMPLE 6 Factor out the greatest common factor from each polynomial. 9y 5 y 2 y 2 9y 3 y 2 1 The greatest common factor is y 2. y 2 9y 3 1 (c) 6 2 t 8t 12t 2t3 2 4 6 14m 4 m 1 28m 3 m 1 7m 2 m 1 The greatest common factor is 7m 2 m 1. Use the distributive property. 14m 4 m 1 28m 3 m 1 7m 2 m 1 7m 2 m 12m 2 4m 1 7m 2 m 12m 2 4m 1 Factoring by Grouping When a polynomial has more than three terms, it can sometimes be factored by a method called factoring by grouping. For eample, to factor collect the terms into two groups so that each group has a common factor. Factor each group, getting a ay 6 6y, a ay 6 6y a ay 6 6y a ay 6 6y a y 6 y. The quantity y is now a common factor, which can be factored out, producing a ay 6 6y ya 6. It is not always obvious which terms should be grouped. Eperience and repeated trials are the most reliable tools for factoring by grouping. EXAMPLE 7 Factor by grouping. mp 2 7m 3p 2 21 mp 2 7m 3p 2 21 m p 2 7 3 p 2 7 p 2 7m 3 Group the terms. Factor each group. p 2 7 is a common factor. 2y 2 2z ay 2 az 2y 2 2z ay 2 az 2y 2 z ay 2 z Factor each group. The epression y 2 z is the negative of y 2 z, so factor out a instead of a. Factoring Trinomials 2y 2 z ay 2 z y 2 z2 a Factor out a. Factor out y 2 z. Factoring is the opposite of multiplying. Since the product of two binomials is usually a trinomial, we can epect factorable trinomials (that have terms with no common factor) to have two binomial factors. Thus, factoring trinomials requires using FOIL backward.
7.6 Polynomials and Factoring 377 EXAMPLE 8 Factor each trinomial. 4y 2 11y 6 To factor this polynomial, we must find integers a, b, c, and d such that 4y 2 11y 6 ay bcy d. By using FOIL, we see that ac 4 and bd 6. The positive factors of 4 are 4 and 1 or 2 and 2. Since the middle term is negative, we consider only negative factors of 6. The possibilities are 2 and 3 or 1 and 6. Now we try various arrangements of these factors until we find one that gives the correct coefficient of y. 2y 12y 6 4y 2 14y 6 2y 22y 3 4y 2 10y 6 y 24y 3 4y 2 11y 6 Incorrect Incorrect Correct The last trial gives the correct factorization. 6p 2 7p 5 Again, we try various possibilities. The positive factors of 6 could be 2 and 3 or 1 and 6. As factors of 5 we have only 1 and 5 or 5 and 1. Try different combinations of these factors until the correct one is found. 2p 53p 1 6p 2 13p 5 3p 52p 1 6p 2 7p 5 Incorrect Correct Thus, 6p 2 7p 5 factors as 3p 52p 1. Each of the special patterns of multiplication given earlier can be used in reverse to get a pattern for factoring. Perfect square trinomials can be factored as follows. Perfect Square Trinomials 2 2y y 2 y 2 2 2y y 2 y 2 EXAMPLE 9 Factor each polynomial. 16p 2 40pq 25q 2 Since 16p 2 4p 2 and 25q 2 5q 2, use the second pattern shown above with 4p replacing and 5q replacing y to obtain 16p 2 40pq 25q 2 4p 2 24p5q 5q 2 4p 5q 2. Make sure that the middle term of the trinomial being factored, 40pq here, is twice the product of the two terms in the binomial 4p 5q. 40pq 24p5q 169 2 104y 2 16y 4 13 4y 2 2, since 2134y 2 104y 2.
378 CHAPTER 7 The Basic Concepts of Algebra y y y y Factoring Binomials The pattern for the product of the sum and difference of two terms gives the following factorization. Difference of Squares 2 y 2 y y A geometric proof for the difference of squares property is shown above. (The proof is only valid for y 0.) 2 y 2 y y y y y Factor out y in the second step. EXAMPLE 10 Factor each of the following polynomials. 4m 2 9 First, recognize that 4m 2 9 is the difference of squares, since 4m 2 2m 2 and 9 3 2. Use the pattern for the difference of squares with 2m replacing and 3 replacing y. Doing this gives 4m 2 9 2m 2 3 2 2m 32m 3. 256k 4 625m 4 Use the difference of squares pattern twice. 256k 4 625m 4 16k 2 2 25m 2 2 16k 2 25m 2 16k 2 25m 2 16k2 25m2 4k 5m4k 5m (c) 2 6 9 y 4 Group the first three terms to obtain a perfect square trinomial. Then use the difference of squares pattern. 2 6 9 y 4 2 6 9 y 4 3 2 y 2 2 3 y 2 3 y 2 3 y 2 3 y 2 Two other special results of factoring are listed below. Each can be verified by multiplying on the right side of the equation. Sum and Difference of Cubes Sum of Cubes Difference of Cubes 3 y 3 y 2 y y 2 3 y 3 y 2 y y 2 EXAMPLE 11 Factor each polynomial. 3 27 Notice that 27 3 3, so the epression is a sum of cubes. Use the first pattern given above. 3 27 3 3 3 3 2 3 9
7.6 Polynomials and Factoring 379 m 3 64n 3 m 3 4n 3 m 4nm 2 m4n 4n 2 m 4nm 2 4mn 16n 2 (c) 8q 6 125p 9 2q 2 3 5p 3 3 2q 2 5p 3 2q 2 2 2q 2 5p 3 5p 3 2 2q 2 5p 3 4q 4 10q 2 p 3 25p 6