Lesson 3: Polynomials and Exponents, Part 1

Transcription

1 Lesson 2: Introduction to Variables Assessment Lesson 3: Polynomials and Exponents, Part 1 When working with algebraic expressions, variables raised to a power play a major role. In this lesson, we look in depth at variables with exponents and how to work with them. We then look at polynomials that contain variables with exponents. Section 3.1: Polynomials Section 3.2: Operations on Polynomials Section 3.3: Properties of Exponents Section 3.4: Multiplication of Polynomials Section 3.5: Applications from Geometry Scottsdale Community College Page 43 Introductory Algebra

2 Lesson 3 Checklist Component Required? Y or N Comments Due Score Online Homework Online Quiz Online Test Practice Problems Lesson Assessment Scottsdale Community College Page 44 Introductory Algebra

3 Name: Date: 3 Section 3.1: Polynomials Definitions Polynomial: An algebraic expression composed of the sum of terms containing a single variable raised to a positive integer exponent. Monomial: A polynomial consisting of one term. Binomial: A polynomial consisting of two terms. Trinomial: A polynomial consisting of three terms. Leading Term: The term that contains the highest power of the variable in a polynomial. Leading Coefficient: The coefficient of the leading term. Degree: The highest exponent in a polynomial. Example 1: Complete the table. Polynomial Name Leading Coefficient Constant Term Degree a + a m + m 2m 8 5x 2 + x 3! 7!2x + 4 4x 3 Scottsdale Community College Page 45 Introductory Algebra

4 You Try 1. Complete the table. Polynomial Name Leading Coefficient Constant Term Degree 3n 2! 2n +8 4x x 2! 7 Scottsdale Community College Page 46 Introductory Algebra

5 Section 3.2: Operations on Polynomials Addition of Polynomials Example 1: Add. ( 3n 2! 2n +8) + ( 3n 3! 7n 2! n! 9) Subtraction of Polynomials Example 2: Subtract. ( a 3 + 5a +11)! ( 4a 3 + 6a 2! a +1) Combine and Simplify Example 3: Perform the indicated operations. Simplify. ( 3x!1)! ( x 2! x! 9) + ( 4x 3 + x 2! 7x + 2) 2. Perform the indicated operations. Simplify. You Try ( 5x +8) + ( 3x 2! 4x!1)! ( 5x 3 + 3x 2! 4x + 6) Scottsdale Community College Page 47 Introductory Algebra

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7 Section 3.3: Properties of Exponents Given any real numbers a, b, c, m, and n n 1 = 1 n = n 0 = 0 n = n! 0 n! 0 Example 1: Evaluate and simplify the following expressions. Assume x 0, x -1/2, a 0, b 0, and c 0. a. 5x 0 b. ( 2x +1) 0 c. a 0 + b 0 + c 0 The Multiplication Property: a m a n = a m+ n Example 2: Simplify the following expressions. a. n 3 n 9 b. b 5! b 4! b c. 5x 2 y 5 ( 7xy 9 ) Raising a Power to a Power: ( m n mn a ) = a Example 3: Simplify the following expressions. a. ( x 3 ) 9 b. 5b 2 ( b 5 ) 8 Scottsdale Community College Page 49 Introductory Algebra

8 Raising a Product to a Power: ( n n ab ) = a b n Example 4: Simplify the following expressions. a. ( 5x) 2 b. ( x 3 y 2 ) 9 c. (!8ab 5 ) 2 d. 5 (!2w 7 ) 3 e. 5n 4 (!3n 3 ) 2 3. Simplify the following expressions. You Try a. 3 (!2x 4 ) 2 b. 2x (!3x 2 ) 3 c. g 3! g 4! g 5 d. 2n 0 Scottsdale Community College Page 50 Introductory Algebra

9 Section 3.4: Multiplication of Polynomials Multiplication of Monomials Example 1: Multiply and simplify. ( 3x 5 )(!2x 9 ) Example 2: Expand and simplify. The Distributive Property 5x 3 ( 2x 5! 4x 3! x +8) Example 3: Multiply and simplify. a. ( x + 3) ( x + 4) Multiplication of Polynomials b. ( m! 5) ( m! 6) c. ( 2d! 4) ( 3d + 5) d. ( x! 2) ( x 2 + 2x! 4) Scottsdale Community College Page 51 Introductory Algebra

10 Example 4: Multiply and simplify. Squaring a Binomial a. ( n + 5) 2 b. ( 3! 2a) 2 4. Multiply and simplify. You Try a.!3x 2 ( x 5 + 6x 3! 5x) b. ( 3x! 4) ( 5x + 2) c. ( 2p! 5) 2 Scottsdale Community College Page 52 Introductory Algebra

11 Section 3.5: Applications from Geometry Example 1: Write a polynomial in simplest form that represents the area of the square. SOLUTION: x y x A C y B D Region A has area: x! x = x 2 Regions B and C each have area: x! y = xy Region D has area: y! y = y 2 Total Area = x 2 + xy + xy + y 2 = y 2 + x 2 + 2xy Example 1 (another way): Write a polynomial in simplest form that represents the area of the square. SOLUTION: x y x A C y B D The total length of each side is x + y. Total Area = ( x + y) ( x + y) = x 2 + xy + yx + y 2 Note that xy and yx are like terms: xy + yx = 2xy Total Area = x 2 + 2xy + y 2 Example 2: Write a polynomial in simplest form that represents the area of the shaded region. SOLUTION: To find the area of the shaded region we find the area of the big square and subtract the area of the little square. The big square has area: y! y = y 2 Each little square has area: x! x = x 2 Area of the shaded region = y 2! 4x 2 Scottsdale Community College Page 53 Introductory Algebra

12 You Try 5. Write a polynomial in simplest form that represents the total area of the figure shown below. x x Write a polynomial in simplest form that represents the area of the dark blue region of the figure shown below. x x 2 5 Scottsdale Community College Page 54 Introductory Algebra