2. SLIDE SHOW INSTRUCTIONS This presentation is completely under your control. This lesson will show only one step at a time, to see the next step you must press a key. to see the next step you must press a key. (Actual names written on a key are in green) TO STOP THE SLIDE SHOW: press ‘escape’ (Esc, top left of keyboard) TO MOVE FORWARD: press the “spacebar” or Enter (PageDn, , , also work) TO MOVE BACKWARD: press the  key (PageUp, or  also work)

3. Polynomial Subtraction: Like Terms

4. CombineLike Terms To subtract polynomials, we must Combine Like Terms Subtract: x 2 - 3x + 4 and 5x 2 - 2x - 2 x 2 - (5x 2 ) - 4x 2 Combine Like Terms by adding or subtracting the coefficients, but keep the variables (and powers) the same. (Use the sign rules of integers to determine whether to add or subtract) These terms both have x 2, so they are like terms Notice that the sign of the second term will change! distribute the negative = x 2 - 5x 2

5. Distribute Negative Sign & CombineLike Terms Distribute Negative Sign & Combine Like Terms Now subtract the next set of like terms: x 2 - 3x + 4 and 5x 2 - 2x - 2 -4x 2 -3x - (-2x) - x These terms both have x, so they are like terms Notice that the sign of the second term will change! distribute the negative = -3x + 2x

6. Distribute Negative Sign & CombineLike Terms Distribute Negative Sign & Combine Like Terms Now subtract the next set of like terms: x 2 - 3x + 4 and 5x 2 - 2x - 2 -4x 2 +4 - (-2) - x Notice that the sign of the second term will change! distribute the negative = 4 + 2 These terms are constants (numbers with no variables), so they are like terms Answer: -4x 2 - x + 6 + 6

7. Horizontal Method

8. Subtract 2x 2 - x - 7 and -x 2 + 3x - 4, use the Horizontal Method: 2x 2 + x 2 3x 2 Combine Like Terms by adding or subtracting the coefficients, but keep the variables (and powers) the same. (Use the sign rules of integers to determine whether to add or subtract) These terms both have x 2, so they are like terms 2x 2 - x - 7 - (-x 2 + 3x - 4) 2x 2 - x - 7 + x 2 - 3x + 4 Notice that all the signs of the second term will change! 2. Distribute the minus sign. 1. Write the two polynomials on one line with a minus sign between them Putting the second polynomial in parentheses helps prevent sign errors

9. Horizontal Method: Horizontal Method: Polynomial Subtraction Now the next set of like terms: -x - 3x 3x 2 - 4x - 4x These terms each have an x, so they are like terms 2x 2 - x - 7 + x 2 - 3x + 4

10. Now the last set of like terms: -7 + 4 3x 2 Answer: 3x 2 - 4x - 3 - 4x - 4x These terms are constants (numbers with no variables), so they are like terms - 3 - 3 2x 2 - x - 7 + x 2 - 3x + 4 Horizontal Method: Horizontal Method: Polynomial Subtraction

11. Practice Problems: (Hit enter to see the answers) Subtract using the Horizontal Method 1) -6x 2 + 2x + 1 and 3x 2 - x + 2 5) 5x + 2x - 3 and 4x + 2 2) 5xy + 4x and -3xy - 12x 6) -3y 2 + 2y and y 2 + y - 1 3) 4ab + 2a 2 b and 3ab 7) 2xy - 5x and - 3xy + 6x - 7 4) 3x 2 y +4x 3 y and - x 3 y + 2x 2 y 8) -17x + 6 and 3x - 6 Answers: 1) -9x 2 + 3x 2) 8xy +16x 3) ab + 2a 2 b 4) x 2 y + 5x 3 y 5) 3x - 5 6) -4y 2 + y + 1 7) 5xy -11x + 7 8) -20x + 12

12. Vertical Method

13. Subtract 4x 2 + 3x - 6 and 2x 2 - 5x + 4, use the Vertical Method : 4x 2 - 2x 2 Write the two polynomials so that the like terms are stacked on top of each other These terms both have x 2, so they are like terms Notice that all the signs in the second polynomial will change!

14. Vertical Method Subtract 4x 2 + 3x - 6 and 2x 2 - 5x + 4, use the Vertical Method : 4x 2 - 2x 2 Write the two polynomials so that the like terms are stacked on top of each other These terms both have an x, so they are like terms Notice that all the signs in the second polynomial will change! + 3x + 5x

15. Vertical Method Subtract 4x 2 + 3x - 6 and 2x 2 - 5x + 4, use the Vertical Method : 4x 2 - 2x 2 Write the two polynomials so that the like terms are stacked on top of each other Notice that all the signs in the second polynomial will change! + 3x + 5x - 6 - 4 These terms are constants (numbers with no variables), so they are like terms

16. Now draw a line under the whole thing and add the coefficients. 4x 2 - 2x 2 + 3x + 5x - 6 - 4 2x 2 + 8x - 10 ANSWER = Vertical Method Subtract 4x 2 + 3x - 6 and 2x 2 - 5x + 4, use the Vertical Method :

17. Vertical Method Subtract x 2 + 2 and 6x 2 - 5x - 3 use the Vertical Method : x 2 - 6x 2 Write the two polynomials so that the like terms are stacked on top of each other These terms both have x 2, so they are like terms Notice that all the signs in the second polynomial will change!

18. Vertical Method x 2 - 6x 2 Notice that all the signs in the second polynomial will change! Subtract x 2 + 2 and 6x 2 - 5x - 3 use the Vertical Method : + 0x + 5x (no x term?) Solution: Write in a zero where there are missing terms. (Or you can leave a blank spot) A problem that comes up when using the Vertical Method is that sometimes there are terms missing.

19. Vertical Method x 2 - 6x 2 Write the two polynomials so that the like terms are stacked on top of each other Notice that all the signs in the second polynomial will change! + 0x + 5x + 2 + 3 These terms are constants (numbers with no variables), so they are like terms Subtract x 2 + 2 and 6x 2 - 5x - 3 use the Vertical Method :

20. Now draw a line under the whole thing and add the coefficients. x 2 - 6x 2 + 0x + 5x + 2 + 3 -5x 2 + 5x + 5 ANSWER = Vertical Method Subtract x 2 + 2 and 6x 2 - 5x - 3 use the Vertical Method :

21. Suggestions for other situations: SituationSolution 1. A term has no coefficient showingWrite a “1” in front of it Example: x 2 + 3x + 1 1x 2 + 3x + 1 2. There are more than two like terms Stack (or group) all like terms together Ex: 2x + 6x - 3 and 4x + 5 (2x + 6x + 4x) + (-3 + 5) 3. There are many missing terms Write in zeros for each of them Ex: 5x 3 - 2x and 4x 4 + 3x 2 + x - 60x 4 + 5x 3 + 0x 2 - 2x + 0 4x 4 + 0x 3 + 3x 2 + 1x - 6 4x 4 + 5x 3 + 3x 2 + 1x - 6

22. Practice Problems: (Hit enter to see the answers) Subtract using the Vertical Method 1)-6x 2 + 2x + 1 and 3x 2 - x + 2 5) 5x + 2x - 3 and 4x + 2 2) 5y + 4x and -3y - 12x 6) -3y 2 + 2y and y 2 + y - 1 3) 4ab + 2a 2 and 3ab 7) 2xy - 5x and -3xy + 6x - 7 4) 3x 2 y + 4xy and - xy + 2x 2 y 8) -17x 2 + 6 and 3x - 6 Answers: 1) -9x 2 + 3x - 1 2) 8y +16x 3) ab + 2a 2 4) x 2 y + 5xy 5) 3x - 5 6) -4y 2 + y + 1 7) 5xy -11x + 7 8) -20x + 12

23. End of Tutorial Go to www.greenebox.comwww.greenebox.com for more great math tutorials for your home computer Questions? send e-mail to: lgreene1@satx.rr.com lgreene1@satx.rr.com