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CHAPTER 5 Polynomials, Polynomial Functions, and Factoring (5.1) Polynomial Functions Polynomials are used in business and science to express various relationships. They are expressed in the form: a n x n a n 1 x n 1 a1 x a 0 They can be classified by either the number of terms, or the highest degree (exponents of the variables). Examples: 1) 5x³ _________________ _________________ 2) x 2 25 _________________ _________________ 3) x 2 5x 7 _________________ _________________ 4) x 3 y 4 3x ² y 3 5 x 4 _________________ _________________ A) EVALUATING POLYNOMIALS Evaluating the expression for the specific values of the variables. f ( x) 2 x² 5x 1 and g ( x) x³ 2 x² 5x 1 a) find f(-2) b) (f – g)(-1) B) ADDITION AND SUBTRACTION OF POLYNOMIALS Remember that we can only combine like terms. Examples: 1) ( x² x) (3x² 2 x 1) 2) (4 x² y xy) (7 x² y 8xy) 3) (5 x 4 y ² 6 x ³ y 7 y ) (3x 4 y ² 5 x ³ y 6 y 8 x) 1 1 3 3 2 1 4) x 4 x ³ x ² 6 x 4 x ³ x ² 6 5 3 8 5 3 2 (5.2) Multiplication of Polynomials When we multiply the terms of polynomials we use the distributive property. 1) multiply the coefficients and add the exponents. 2) Multiplying binomials we use the procedure known as FOIL. Examples: 1) 6 x ³(3x5 5 x ² 7) 2) 3x ² y (10 x ² y 4 2 xy ³ 7) 3) ( x 1)(x² x 1) 4) (2 x 3)(6 x 4) 5) (3xy 1)(5xy 2) 6) (3x 4 y )² 7) [(3x 2) 5 y][(3x 2) 5 y] 8. (7 x 5 y 2)(7 x 5 y 2) (5.3-5.6) Factoring STEPS FOR FACTORING: Factoring polynomials is the reverse process of multiplication of polynomials. Factoring polynomials will aid us in solving equations where we use the principle of zeros products. Use the following steps to aid in the factoring of polynomials: 1) Arrange the terms in descending order according to the exponents 2) Factor out any common factors – reverse of the distributive property ac + ad = a(c + d). (The leading coefficient should be positive) 3) Factor the resulting polynomial: 4 terms – try the grouping method (grouping 2 terms and factoring each) 3 terms – determine the 2 binomials 2 terms – the following properties may apply: a² - b² = (a + b)(a – b) a³ + b³ = (a + b)(a² - ab + b²) a³ - b³ = (a = b)(a² + ab + b²) How can you check your factoring?_______________________________________ FACTORING OUT COMMON FACTORS: Look for factors common to every term in an expression and then use the reverse of the distributive law: ac + ad = a(c + d) Examples: Factor 1) 6y² + 3y 2) 5x² - 5x + 15 3) -2x² + 12x + 40 4) 25a 3b 5 50a 4 b 4 15a 5 b² 5) x(y – 9) – 5(y – 9) THE GROUPING METHOD: This method may be applied to polynomials with 4 terms. Not every 4 term polynomial is factorable using this method. 1. Group two terms together 2. Factor each group (the parentheses must be the same to continue) 3. Factor again into 2 binomials Examples: 1) x² + 2x + 4x + 8 2) xy – 5x + 9y - 45 3) x³ 2x² 5x 10 4) 3x³ - 2x² - 6x + 4 FACTORING TRINOMIALS: When we factor trinomials we are finding factors that will multiply out to give the initial polynomial. There are several methods one can use to do this. There is the method of writing all trinomials into four terms so the grouping method can be used to factor or one may feel comfortable with trial and error. At this point, the trial and error method should be used. Examples: Factor the following: 1. x² + 6x + 5 2. t² - 15 – 2t 3. 2a² - 16a + 32 4. 3x² - 16x – 12 5. 4y³ + 12y² - 72y 6. 10x³ + 24x² + 14x FACTORING PERFECT SQUARES AND CUBES When we have a polynomial with 2 terms to factor, look for the following: a² - b² = (a + b)(a – b) a² + b² = does not factor a³ + b³ = (a + b)(a² - ab + b²) a³ - b³ = (a – b)(a² + ab + b²) Examples: 1) 4x² - 9 2) 49m² 100 p ² 3) x²y² - 100 4) (x – 6)² - y² 5) 2x³y – 18xy 6) x³ + 3x² - 9x – 27 7) x4 6 x² 9 8) 16x² - 40xy + 25y² 9) x³ - 27 10) 27x³ + 8 (5.7) Solving Quadratic Equations by Factoring When we solve polynomials, we are looking for the solution to the equation. Basically, we are looking for the value or values of the variable that will make the equation true. This means that all answers can be checked. Solving a quadratic equation by factoring: 1) Bring all terms to one side of the equation so the equation equals zero. 2) Factor the polynomial 3) Using the Zero Factor Property, set each factor to zero. 4) Solve each resulting equation for the variable. 5) Check your solution ZERO FACTOR PROPERTY: For any real numbers a and b, if ab = 0 then a = 0 or b = 0 If a = 0 or b = 0 then ab = 0 Examples: Solve each equation: 1) t² - 3t – 28 = 0 2) 5x² = 8x - 3 3) 2x² = 5x 4) 4x² = 25 x² 5x 5) (x – 1)(x + 4) = 14 6) 60 4 2 7) x³ + 2x² = 16x + 32 8) 3x4 81x APPLICATIONS: 1. An envelope is 4 cm longer than it is wide. The area is 96 cm². Find the length and the width. 2. A tool box is 2 ft high, and its width is 3 ft less than its length. If its volume is 80 ft³, find the length and the width of the box. 3. A tree is supported by a wire anchored in the ground 5 feet from its base. The wire is 1 foot longer than the height that it reaches on the tree. Find the length of the wire.