Name Class Date 6-4 Adding and Subtracting Polynomials Going Deeper Essential question: How do you add and subtract polynomials? To add or subtract polynomials, you combine like terms. You can add or subtract horizontally or vertically. 1 Add. A-APR.1.1 EXAMPLE Adding Polynomials A (4 x 3 + 12 x 2 + 8x + 6) + (5 x 2-6x + 9) Use a vertical arrangement. 4 x 3 + 12 x 2 + 8x + 6 Write the polynomials, aligning like terms. 5 x 2-6x + 9 x 3 + x 2 + x + Add the coefficients of like terms. B (2x - 7 x 2 ) + ( x 2-2x + 5) Use a horizontal arrangement. (-7 x 2 + 2x) + ( x 2-2x + 5) Write the polynomials in standard form. = (-7 x 2 + ) + (2x - ) + Group like terms. = + 0x + Add the coefficients of like terms. = + Simplify. REFLECT 1a. Do you get the same results whether you add polynomials vertically or horizontally? Why or why not? 1b. Is the sum of two polynomials always another polynomial? Explain. 1c. Is the sum of two polynomials of degree 5 always a polynomial of degree 5? Give an example to explain your answer. Chapter 6 343 Lesson 4
To subtract polynomials, you add the opposite of the subtracted polynomial. The following example shows how to use this method with the vertical and horizontal formats. 2 A-APR.1.1 EXAMPLE Subtract. Subtracting Polynomials A (2 + 9 x 2 ) - (-6 x 2-3x + 1) Use a vertical arrangement. 9 x 2 + 2 Write the first polynomial in standard form. 6 x 2 + 3x - 1 Add the opposite of the second polynomial. x 2 + x + Add the coefficients of like terms. B (6 x 3 + 3 x 2 + 2x + 9) - (4 x 3 + 6 x 2-2x + 7) Use a horizontal arrangement. (6 x 3 + 3 x 2 + 2x + 9) - (4 x 3 + 6 x 2-2x + 7) Write the polynomials. = (6 x 3 + 3 x 2 + 2x + 9) + (-4 x 3-6 x 2 + 2x - 7) Add the opposite. = (6 x 3 - ) + (3 x 2 - ) + ( + 2x) + ( - 7) Group like terms. = x 3 - x 2 + x + Add the coefficients of like terms. REFLECT 2a. How is subtracting polynomials similar to subtracting integers? 2b. In part A, you leave a gap in the polynomial 9 x 2 + 2 when you write the subtraction problem vertically. Why? 2c. Is the difference of two polynomials always another polynomial? Explain. Chapter 6 344 Lesson 4
3 F-BF.1.1a EXAMPLE Modeling High School Populations According to data from the U.S. Census Bureau for the period 2000 2007, the number of male students enrolled in high school in the United States can be approximated by the function M(x) = -0.004 x 3 + 0.037 x 2 + 0.049x + 8.11 where x is the number of years since 2000 and M(x) is the number of male students in millions. The number of female students enrolled in high school in the United States can be approximated by the function F(x) = 0.006 x 3 + 0.029 x 2 + 0.165x + 7.67 where x is the number of years since 2000 and F(x) is the number of female students in millions. Estimate the total number of students enrolled in high school in the United States in 2007. A Make a plan. The problem asks for the total number of students in 2007. First find T(x) = M(x) + F(x) to find a model for the total enrollment. Then evaluate T(x) at an appropriate value of x to find the total enrollment in 2007. B Add the polynomials. -0.004 x 3 + 0.037 x 2 + 0.049x + 8.11 Write the polynomials, aligning like terms. -0.006 x 3 + 0.029 x 2 + 0.165x + 7.67 x 3 + x 2 + x + Add the coefficients of like terms. T(x) = C Evaluate T (x). For 2007, x = 7. Use a calculator to evaluate T (7). Round to one decimal place. T (7) So, there were approximately high school students in 2007. REFLECT 3a. Is it possible to solve this problem without adding the polynomials? Explain. 3b. Explain how you can use the given information to estimate how many more male high school students than female high school students there were in the United States in 2007. Chapter 6 345 Lesson 4
PRACTICE Add or subtract. 1. (2 x 4-6 x 2 + 8) + (- x 4 + 3 x 2-12) 2. (7 x 2-2x + 1) + (8 x 3 + 2 x 2 + 7x - 4) 3. (5 x 2-6 x 3 + 11) + (9 x 3 + 3x + 7 x 4 ) 4. (-3 x 3-7 x 5-3) + (5 x 2 + 3 x 3 + 7 x 5 ) 5. (2 x 4-6 x 2 + 8) - (- x 4 + 5 x 2-12) 6. ( x 3 + 25) - (- x 2-18x - 12) 7. (2 x 2 + 3x + 1) - (7 x 2-2x + 7 x 3 ) 8. (10 x 2 + 3) - (15 x 2-4x + 9 x 4 + 7) 9. (14 x 4 - x 3 + 2 x 2 + 5x + 15) - (10 x 4 + 3 x 3-5 x 2-6x + 4) 10. (-6 x 3 + 10x + 26) + (5 x 2-6 x 5 + 7x) + (3-22 x 4 ) 11. According to data from the U.S. Census Bureau, the total number of people in the United States labor force can be approximated by the function T(x) = -0.011 x 2 + 2x + 107, where x is the number of years since 1980 and T(x) is the number of workers in millions. The number of women in the United States labor force can be approximated by the function W(x) = -0.012 x 2 + 1.26x + 45.5. a. Write a polynomial function M(x) that models the number of men in the labor force. b. Estimate the number of men in the labor force in 2008. Explain how you made your estimate. 12. Error Analysis A student was asked to find the difference (4 x 5-3 x 4 + 6 x 2 ) - (7 x 5-6 x 4 + x 3 ). The student s work is shown at right. Identify the student s error and give the correct difference. 4 x 5-3 x 4 + 6 x 2-7 x 5-6 x 4 + x 3-3 x 5-9 x 4 + x 3 + 6 x 2 Chapter 6 346 Lesson 4
Name Class Date 6-4 Additional Practice Add or subtract. 1. 3 3 8 3 3 3 2 2 2. 2 5 12 5 6 5 Add. 3. 3 2 2 7 4. 5 2 2 3 5. 11 3 3 2 8 2 6 2 5 6 9 3 2 3 6. ( 2 13 4 ) (3 2 7 ) 7. (4 3 2 4 ) ( 3 2 4 ) Subtract. 8. 12 2 3 9. 2 5 3 4 8 10. 4 6 2 ( 4 2 2 8 ) (3 5 2 4 8) ( 6 4 2 2 ) 11. ( 2 8 ) ( 12 2 2 8 ) 12. ( 2 2 3 ) (3 3 2 4 ) 13. Antoine is making a banner in the shape of a triangle. He wants to line the banner with a decorative border. How long will the border be? 14. Darnell and Stephanie have competing refreshment stand businesses. Darnell s profit can be modeled with the polynomial 2 8 100, where is the number of items sold. Stephanie s profit can be modeled with the polynomial 2 2 7 200. a. Write a polynomial that represents the difference between Stephanie s profit and Darnell s profit. b. Write a polynomial to show how much they can expect to earn if they decided to combine their businesses. Chapter 6 347 Lesson 4
Problem Solving 1. There are two boxes in a storage unit. The volume of the first box is 4 3 4 2 cubic units. The volume of the second box is 6 3 18 2 cubic units. Write a polynomial for the total volume of the two boxes. 3. Two cabins on opposite banks of a river are 12 2 7 5 feet apart. One cabin is 9 1 feet from the river. The other cabin is 3 2 4 feet from the river. Write the polynomial that represents the width of the river where it passes between the two cabins. Then calculate the width if 3. 2. The recreation field at a middle school is shaped like a rectangle with a length of 15 yards and a width of 10 3 yards. Write a polynomial for the perimeter of the field. Then calculate the perimeter if 2. 4. The angle value of Greg s sector can be modeled by 2 6 2. The angle value of Dion s sector can be modeled by 7 20. Which polynomial represents both sectors combined? A 2 18 C 6 2 7 18 B 2 13 22 D 7 2 6 22 5. The sum of Greg and Lynn s sectors is 2 2 4 6. The sum of Max and Dion s sectors is 10 26. Which polynomial represents how much greater Greg and Lynn s combined sectors are than Max and Dion s? F 2 2 6 32 H 2 2 6 32 G 2 2 6 20 J 2 2 14 20 6. The sum of Lynn s sector and Max s sector is 2 2 9 2. Max s sector can be modeled by 3 6. Which polynomial represents the angle value of Lynn s sector? A 2 2 6 4 C 2 2 12 8 B 2 2 6 4 D 2 2 12 8 Chapter 6 348 Lesson 4