1. Warm-up State whether each expression is a polynomial. If the expession is a polynomial, identify it as either a monomial, binomial, or trinomial and give its degree. 1. 8a 2 + 5ab 2. 3x 2 + 4x – 7/x 3. 5x 2 + 7x + 2 4. 6a 2 b 2 + 7ab 5 – 6b 3 5. w 2 x - + 6x 7w 3

2. Warm-up State whether each expression is a polynomial. If the expession is a polynomial, identify it as either a monomial, binomial, or trinomial and give its degree. 1. 8a 2 + 5ab binomial; 2 2. 3x 2 + 4x – 7/xnot a polynomial not the product of # and variable 3. 5x 2 + 7x + 2trinomial; 2 4. 6a 2 b 2 + 7ab 5 – 6b 3 trinomial; 2 5. w 2 x - + 6xtrinomial; 2 7w 3

3. Homework 6.5 1. 2 2. 3 3. 4 4. 1 5. 4 6. 4 7. 4 8. 8 9. 4 10. 5 11. 5 + 2x 2 + 3x 3 + x 4 12. 1 + 3x + 2x 2 13. - 6 + 5x + 3x 2 14. 2 + x + 9x 2 + x 3 15. - 3 + 4x – x 2 + 3x 3 16. – 2x + x 2 - x 3 + x 4

4. Homework 6.5 17. 6 + 12x + 6x 2 + x 3 18. 21r 2 + 7r 5 x – r 2 x 2 – 15x 3 19. 5x 3 - 3x 2 + x + 4 20. - x 3 + x 2 - x + 1 21. 3x 3 + x 2 - x + 27 22. 3x 3 + x 2 + x - 17 23. x 3 + x - 1 24. 3x 3 + x 2 - x + 64

5. Homework 6.5 25. - x 3 + x + 25 26. ⅓p x 3 + p 3 x 2 + px + 5p

6. 6.6 Adding and Subtracting Polynomials CORD Math Mrs. Spitz Fall 2006

7. Objectives: After studying this lesson, you should be able to add and subtract polynomials. After studying this lesson, you should be able to add and subtract polynomials.

8. Assignment: 6.6 Worksheet 6.6 Worksheet

9. Application: The standard measurement for a window is the united inch. The united inch measurement of a window is equal to the sum of the length of the length and the width of the window. If the length of the window at the right is 2x + 8 and the width is x – 3 inches, what is the size of the window in united inches? The standard measurement for a window is the united inch. The united inch measurement of a window is equal to the sum of the length of the length and the width of the window. If the length of the window at the right is 2x + 8 and the width is x – 3 inches, what is the size of the window in united inches? x – 3 in.

10. Application: The size of the window is (2x + 8) + (x – 3) inches. To add two polynomials, add the like terms. The size of the window is (2x + 8) + (x – 3) inches. To add two polynomials, add the like terms. = (2x +8) + (x - 3) = (2x +8) + (x - 3) = 2x + 8 + x – 3 = 2x + 8 + x – 3 = (2x + x) + (8 – 3) = (2x + x) + (8 – 3) = 3x + 5 = 3x + 5 The size of the window in united inches is 3x + 5 inches. x – 3 in.

11. Application You can add polynomials by grouping the like terms together and then finding the sum (as in the example previous), or by writing them in column form. You can add polynomials by grouping the like terms together and then finding the sum (as in the example previous), or by writing them in column form.

12. Example 1: Find (3y 2 + 5y – 6) + (7y 2 -9) Method 1: Group the like terms together. (3y 2 + 5y – 6) + (7y 2 -9) = (3y 2 + 7y 2 ) + 5y + [-6 + (-9)] = (3 + 7)y 2 + 5y + (-15) = 10y 2 + 5y - 15

13. Example 2: Find (3y 2 + 5y – 6) + (7y 2 -9) Method 2: Column form 3y 2 5y – 6 + 7y 2 – 9 10y 2 5y – 15 Recall that you can subtract a rational number by adding its additive inverse or opposite. Similarly, you can subtract a polynomial by adding its additive inverse.

14. To find the additive inverse of a polynomial, replace each term with its additive inverse. Polynomial Additive Inverse x + 2y -x – 2y 2x 2 – 3x +5 - 2x 2 + 3x -5 - 8x + 5y – 7z 8x - 5y + 7z 3x 3 - 2x 2 – 5x - 3x 3 + 2x 2 + 5x The additive inverse of every term must be found!!!

15. Example 2: Find (4x 2 – 3y 2 + 5xy) – (8xy+ 6x 2 + 3y 2 ) Method 1: Group the like terms together. (4x 2 – 3y 2 + 5xy) – (8xy+ 6x 2 + 3y 2 ) (4x 2 – 3y 2 + 5xy) – (8xy+ 6x 2 + 3y 2 ) = (4x 2 – 3y 2 + 5xy) + (– 8xy - 6x 2 - 3y 2 ) = (4x 2 - 6x 2 ) + (5xy – 8xy) + (- 3y 2 - 3y 2 ) = (4 - 6)x 2 + (5 – 8)xy + (-3 - 3)y 2 = -2x 2 – 3xy + -6y 2 OR WOULD YOU PREFER COLUMN FORMAT?

16. Example 2: Find (4x 2 – 3y 2 + 5xy) – (8xy+ 6x 2 + 3y 2 ) Column format 4x 2 5xy -3y 2 - 6x 2 8xy 3y 2 First, reorder the terms so that the powers of x are in descending order: (4x 2 + 5xy – 3y 2 ) – (6x 2 + 8xy+ 3y 2 ) THEN use the additive inverse to change the signs

17. Example 2: Find (4x 2 – 3y 2 + 5xy) – (8xy+ 6x 2 + 3y 2 ) Column format 4x 2 5xy -3y 2 + - 6x 2 - 8xy - 3y 2 - 2x 2 - 3xy - 6y 2 -2x 2 – 3xy + -6y 2 and 6x 2 + 8xy+ 3y 2 To check this result, add -2x 2 – 3xy + -6y 2 and 6x 2 + 8xy+ 3y 2 (4x 2 + 5xy – 3y 2 ) This is what you should get after you check it.

18. Example 3: Find the measure of the third side of the triangle. P is the measure of the perimeter. The perimeter is the sum of the measures of the three sides of the triangle. Let s represent the measure of the third side. The perimeter is the sum of the measures of the three sides of the triangle. Let s represent the measure of the third side. 8x 2 – 8x + 5 3x 2 + 2x - 1 s P = 12x 2 – 7x + 9

19. (12x 2 – 7x + 9) = (3x 2 + 2x - 1) + (8x 2 – 8x + 5) + s (12x 2 – 7x + 9) - (3x 2 + 2x - 1) - (8x 2 – 8x + 5) = s 12x 2 – 7x + 9 - 3x 2 - 2x + 1 - 8x 2 + 8x - 5) = s (12x 2 - 3x 2 - 8x 2 )+(– 7x - 2x + 8x) + (9 + 1 - 5) = s x 2 - x + 5 = s The measure of the third side is x 2 - x + 5.