333371_0P03.qxp 12/27/06 9:31 AM Page 24 24 Chapter P Prerequisites P.3 Polynomials and Factoring Polynomials What you should learn ᭿ Write polynomials in standard form. An algebraic expression is a collection of variables and real numbers. The most ᭿ Add,subtract,and multiply polynomials. polynomial. common type of algebraic expression is the Some examples are ᭿ Use special products to multiply polyno- 2x ϩ 5, 3x 4 Ϫ 7x 2 ϩ 2x ϩ 4, and 5x 2y 2 Ϫ xy ϩ 3. mials. ᭿ Remove common factors from polynomi- The first two are polynomials in x and the third is a polynomial in x and y. The als. terms of a polynomial in x have the form ax k, where a is the coefficient and k is ᭿ Factor special polynomial forms. the degree of the term. For instance, the polynomial ᭿ Factor trinomials as the product of two binomials. 3 Ϫ 2 ϩ ϭ 3 ϩ ͑Ϫ ͒ 2 ϩ ͑ ͒ ϩ 2x 5x 1 2x 5 x 0 x 1 ᭿ Factor by grouping. has coefficients 2,Ϫ5, 0, and 1. Why you should learn it Polynomials can be used to model and solve Definition of a Polynomial in x real-life problems.For instance,in Exercise 157 on page 34,a polynomial is used to Let a0, a1, a2, . , an be real numbers and let n be a nonnegative integer. model the total distance an automobile A polynomial in x isP.3 an expression of the form travels when stopping. n ϩ nϪ1 ϩ . ϩ ϩ an x anϪ1x a1x a 0 where an 0. The polynomial is of degree n, an is the leading coeffi- cient, and a0 is the constant term. In standard form, a polynomial in x is written with descending powers of x. Polynomials with one, two, and three terms are called monomials, binomials, and trinomials, respectively. A polynomial that has all zero coefficients is called the zero polynomial, denoted by 0. No degree is assigned to this particular polynomial. For polyno- mials in more than one variable, the degree of a term is the sum of the exponents of the variables in the term. The degree of the polynomial is the highest degree of its terms. For instance, the degree of the polynomial Ϫ2x3y6 ϩ 4xy Ϫ x7y4 is 11 because the sum of the exponents in the last term is the greatest. Expressions © Robert W. Ginn/age fotostock such as the following are not polynomials. 3 3 1͞2 x Ϫ Ί3x ϭ x Ϫ ͑3x͒ The exponent 1͞2 is not an integer. 5 Ϫ x2 ϩ ϭ x2 ϩ 5x 1 The exponent Ϫ1 is not a nonnegative integer. STUDY TIP x Expressions are not polynomials Example 1 Writing Polynomials in Standard Form if: Polynomial Standard Form Degree 1. A variable is underneath a radical. a. 4x 2 Ϫ 5x 7 Ϫ 2 ϩ 3x Ϫ5x 7 ϩ 4x 2 ϩ 3x Ϫ 2 7 2. A polynomial expression b. 4 Ϫ 9x 2 Ϫ9x 2 ϩ 4 2 (with degree greater than 0) c. 88͑8 ϭ 8x 0͒ 0 is in the denominator of a Now try Exercise 15. term. 333371_0P03.qxp 12/27/06 9:31 AM Page 25 Section P.3 Polynomials and Factoring 25 Operations with Polynomials You can add and subtract polynomials in much the same way you add and sub- tract real numbers. Simply add or subtract the like terms (terms having exactly the same variables to exactly the same powers) by adding their coefficients. For instance,Ϫ3xy 2 and 5xy2 are like terms and their sum is Ϫ3xy2 ϩ 5xy2 ϭ ͑Ϫ3 ϩ 5͒xy2 ϭ 2xy2. Example 2 Sums and Differences of Polynomials STUDY TIP Perform the indicated operation. When a negative sign precedes a. ͑5x 3 Ϫ 7x 2 Ϫ 3͒ ϩ ͑x 3 ϩ 2x 2 Ϫ x ϩ 8͒ an expression within parentheses, ͑ 4 Ϫ 2 Ϫ ϩ ͒ Ϫ ͑ 4 Ϫ 2 ϩ ͒ b. 7x x 4x 2 3x 4x 3x treat it like the coefficient ͑Ϫ1͒ and distribute the negative sign Solution to each term inside the a. ͑5x 3 Ϫ 7x 2 Ϫ 3͒ ϩ ͑x 3 ϩ 2x 2 Ϫ x ϩ 8͒ parentheses. ϭ ͑5x 3 ϩ x 3͒ ϩ ͑Ϫ7x 2 ϩ 2x 2͒ Ϫ x ϩ ͑Ϫ3 ϩ 8͒ Group like terms. Ϫ͑x2 Ϫ x ϩ 3͒ ϭϪx2 ϩ x Ϫ 3 ϭ 6x 3 Ϫ 5x 2 Ϫ x ϩ 5 Combine like terms. b. ͑7x 4 Ϫ x2 Ϫ 4x ϩ 2͒ Ϫ ͑3x 4 Ϫ 4x2 ϩ 3x͒ ϭ 7x 4 Ϫ x2 Ϫ 4x ϩ 2 Ϫ 3x 4 ϩ 4x2 Ϫ 3x Distributive Property ϭ ͑7x 4 Ϫ 3x 4͒ ϩ ͑Ϫx2 ϩ 4x2͒ ϩ ͑Ϫ4x Ϫ 3x͒ ϩ 2 Group like terms. ϭ 4x 4 ϩ 3x2 Ϫ 7x ϩ 2 Combine like terms. Now try Exercise 23. To find the product of two polynomials, use the left and right Distributive Properties. Example 3 Multiplying Polynomials: The FOIL Method When using the FOIL method, the following scheme may be helpful. ͑3x Ϫ 2͒͑5x ϩ 7͒ ϭ 3x͑5x ϩ 7͒ Ϫ 2͑5x ϩ 7͒ F ϭ ͑3x͒͑5x͒ ϩ ͑3x͒͑7͒ Ϫ ͑2͒͑5x͒ Ϫ ͑2͒͑7͒ L 2 ϭ 15x ϩ 21x Ϫ 10x Ϫ 14 ͑3x Ϫ 2͒͑5x ϩ 7͒ I Product of Product of Product of Product of O First terms Outer terms Inner terms Last terms ϭ 15x 2 ϩ 11x Ϫ 14 Note that when using the FOIL Method (which can be used only to multiply two binomials), the outer (O) and inner (I) terms may be like terms that can be combined into one term. Now try Exercise 39. 333371_0P03.qxp 12/27/06 9:31 AM Page 26 26 Chapter P Prerequisites Example 4 The Product of Two Trinomials Find the product of 4x2 ϩ x Ϫ 2 and Ϫx2 ϩ 3x ϩ 5. Solution When multiplying two polynomials, be sure to multiply each term of one polynomial by each term of the other. A vertical format is helpful. 4 x2 ϩ x Ϫ 2 Write in standard form. ϫ Ϫx2 ϩ 3x ϩ 5 Write in standard form. 2 0 x2 ϩ 5x Ϫ 10 5͑4x2 ϩ x Ϫ 2͒ 1 2 x3 ϩ 3x2 Ϫ 6x 3x͑4x2 ϩ x Ϫ 2͒ Ϫ4x4 Ϫ x3 ϩ 2x2 Ϫx2͑4x2 ϩ x Ϫ 2͒ Ϫ4x4 ϩ 11x3 ϩ 25x2 Ϫ x Ϫ 10 Combine like terms. Now try Exercise 59. Special Products Special Products Let u and v be real numbers, variables, or algebraic expressions. Special Product Example Sum and Difference of Same Terms ͑u ϩ v͒͑u Ϫ v͒ ϭ u 2 Ϫ v 2 ͑x ϩ 4͒͑x Ϫ 4͒ ϭ x 2 Ϫ 42 ϭ x2 Ϫ 16 Square of a Binomial ͑u ϩ v͒2 ϭ u 2 ϩ 2uv ϩ v 2 ͑x ϩ 3͒2 ϭ x 2 ϩ 2͑x͒͑3͒ ϩ 32 ϭ x2 ϩ 6x ϩ 9 ͑u Ϫ v͒2 ϭ u 2 Ϫ 2uv ϩ v 2 ͑3x Ϫ 2͒2 ϭ ͑3x͒2 Ϫ 2͑3x͒͑2͒ ϩ 22 ϭ 9x2 Ϫ 12x ϩ 4 Cube of a Binomial ͑u ϩ v͒3 ϭ u 3 ϩ 3u 2v ϩ 3uv 2 ϩ v 3 ͑x ϩ 2͒3 ϭ x 3 ϩ 3x 2͑2͒ ϩ 3x͑22͒ ϩ 23 ϭ x3 ϩ 6x2 ϩ 12x ϩ 8 ͑u Ϫ v͒3 ϭ u 3 Ϫ 3u 2v ϩ 3uv 2 Ϫ v 3 ͑x Ϫ 1͒3 ϭ x 3 Ϫ 3x 2͑1͒ ϩ 3x͑12͒ Ϫ 13 ϭ x3 Ϫ 3x2 ϩ 3x Ϫ 1 Example 5 The Product of Two Trinomials To understand the individual patterns of special products, have students “derive” Find the product of x ϩ y Ϫ 2 and x ϩ y ϩ 2. each product. Then explain how special products save time and how the pattern of Solution the product must be recognized to factor By grouping x ϩ y in parentheses, you can write the product of the trinomials an expression. as a special product. ͑x ϩ y Ϫ 2͒͑x ϩ y ϩ 2͒ ϭ ͓͑x ϩ y͒ Ϫ 2͔͓͑x ϩ y͒ ϩ 2͔ ϭ ͑x ϩ y͒ 2 Ϫ 22 ϭ x 2 ϩ 2xy ϩ y 2 Ϫ 4 Now try Exercise 61. 333371_0P03.qxp 12/27/06 9:31 AM Page 27 Section P.3 Polynomials and Factoring 27 Factoring The process of writing a polynomial as a product is called factoring. It is an important tool for solving equations and for simplifying rational expressions. Unless noted otherwise, when you are asked to factor a polynomial, you can assume that you are looking for factors with integer coefficients. If a polynomial cannot be factored using integer coefficients, it is prime or irreducible over the integers. For instance, the polynomial x 2 Ϫ 3 is irreducible over the integers. Over the real numbers, this polynomial can be factored as x 2 Ϫ 3 ϭ ͑x ϩ Ί3͒͑x Ϫ Ί3͒. A polynomial is completely factored when each of its factors is prime. So, x 3 Ϫ x 2 ϩ 4x Ϫ 4 ϭ ͑x Ϫ 1͒͑x 2 ϩ 4͒ Completely factored Activities is completely factored, but 1. Explain what happens when the paren- theses are removed from the expression: 3 Ϫ 2 Ϫ ϩ ϭ ͑ Ϫ ͒͑ 2 Ϫ ͒ Not completely factored x x 4x 4 x 1 x 4 Ϫ͑3x2 Ϫ 4x ϩ 2͒. is not completely factored. Its complete factorization is Answer: The sign of each term changes. Ϫ3x2 ϩ 4x Ϫ 2 x 3 Ϫ x 2 Ϫ 4x ϩ 4 ϭ ͑x Ϫ 1͒͑x ϩ 2͒͑x Ϫ 2͒.