1. For Educational Use Only © 2010 10.4 Solving Polynomial Equations in Factored Form Brian Preston Algebra 1 2009-2010

2. For Educational Use Only © 2010 Real World Application How high is the tallest arch in the world?

3. For Educational Use Only © 2010 Lesson Objectives 1) Solve a polynomial equation in factored form. 2) Relate factors and x-intercepts.

4. For Educational Use Only © 2010 Review (3x + 4)(x + 5)5 x5x 4 4 3x 1) 3x 2 + 4x + 15x+ 20 3x 2 + 19x + 20 First Outside Inside Last

5. For Educational Use Only © 2010 (z – 4) What does this mean? Review (z – 4) 2 There are two (z- 4)

6. For Educational Use Only © 2010 – 3 + 2 (x – 2) (x + 3) 2) Definition Starting with 2 or more binomials and solving for the variable. = 0 (x + 3) = 0 x + 3 x – 2 x + 2 = 2 ( ) ( ) x – 3 =

7. For Educational Use Only © 2010 aa b Rule Zero-Product Property = the product of two factors is zero only when at least one of the factors is zero. = 0 b b a a = 0 ( ) ( ) b = 0 or

8. For Educational Use Only © 2010 3) An arch is modeled by the equations y = – 0.00635(x – 315)(x + 315) with x & y measured in feet. How wide is the base of the arch? How high is the arch? wide high y = – 0.00635(x – 315)(x + 315) Real World Application

9. For Educational Use Only © 2010 – 0.00635(x – 315)(x + 315) –315 +315 All you need to graph are the x-intercepts & the vertex. (x – 315) (x + 315) – 0.00635 Example = 0 x + 315 x - 315 x +315 = 315 ( ) ( ) x –315 = = 0 –0.00635( ) y = 3)

10. For Educational Use Only © 2010 Definition All you need to graph are the x-intercepts & the vertex. x-intercepts = 315 & – 315 vertex = (Average of the x-int, #) – 0.00635(x – 315)(x + 315) = 0 y = 3)

11. For Educational Use Only © 2010 630 00 – 0.00635(x – 315)(x + 315) 3) Definition = 0 y = x-intercepts = 315 & – 315 vertex = (Average of the x-int, #) 315 + – 315 2 = 0, (? ) – 0.00635( – 315) ( + 315) = 630 y = – 315 315 0 0x x

12. For Educational Use Only © 2010 x-intercepts 3) y = – 0.00635(x – 315)(x + 315) -200 -600 -1000 -200 -600 200 600 1000 200 600 1000 -1000 (-315,0) (315,0) (0,630) Review vertex 315 & – 315 (0,630)

13. For Educational Use Only © 2010 Real World Application How high is the tallest arch in the world? 630 feet

14. For Educational Use Only © 2010 (x + 5) 4) Example Solve. = 0 2

15. For Educational Use Only © 2010 – 5 (x + 5) 4) Example Solve. = 0 x + 5 x – 5 = ( ) ( ) x =

16. For Educational Use Only © 2010 (4c – 8) (7c + 21) (4c – 8) (7c + 21) 7 7 – 21 4 4 + 8 5) Example Solve. = 0 7c + 21 4c – 8 4c + 8 = 8 ( ) ( ) 7c – 21 = + 0 4 c=2 7 c=– 3

17. For Educational Use Only © 2010 – 1 – 4 – 1 – 4 (x + 1) (x + 4) (x + 1) 7 7 6) Example Solve. = 0 x + 1 x + 4 x – 4 = ( ) ( ) x – 1 = = 0 7( )

18. For Educational Use Only © 2010 (b + 6) 7) Example Solve. = 0 2 8 (b – 9)

19. For Educational Use Only © 2010 – 6 + 9 (b + 6) (b – 9) (b + 6) (b – 9) Example Solve. = 0 b – 9 b + 9 = 9 ( ) 7) 8 = 0 8( ) b + 6 b – 6 = ( ) = 0 b + 6 b – 6 = ( ) 8

20. For Educational Use Only © 2010 (x – 2) (x + 3) Definition These equations can be graphed. = 0 y = The variable solutions are x-intercepts. They allow you to graph more easily.

21. For Educational Use Only © 2010 1 11 11 8) Sketch the graph of y = 2 – 2 – 3. b 2(1) – (– 2) a = = 2 2 – 2 1 1 x x (1) 0 2 3 – 1 x y Review Before, you had to do this to graph.

22. For Educational Use Only © 2010 – 4 8) Sketch the graph of y = 1(1) 2 – 2(1) – 3. = y 1 – 2 = y = – 3 1 – 5 – 4 01 2 3 – 1 – 4 x y (1) Review

23. For Educational Use Only © 2010 – 3 2 22 8) Sketch the graph of y = 2 – 2 – 3. = y 1 – 4 = y = – 3 4 – 7 x (2)(2) x (2)(2) 1 – 4 – 3 01 3 – 1 x y (4) Review

24. For Educational Use Only © 2010 0 3 0 2 33 8) Sketch the graph of y = 2 – 2 – 3. = y 1 – 6 = y = – 3 9 – 9 x (3)(3) x (3)(3) 1 – 4 – 3 0 01 – 1 x y (9) Review

25. For Educational Use Only © 2010 0 00 – 3 3 x y 2 8) Sketch the graph of y = 2 – 2 – 3. = y 1 – 0 = y = – 3 0 x (0)(0) x (0)(0) 1 – 4 – 3 0 1 – 1 (0) Review

26. For Educational Use Only © 2010 – 1 0 0 0 3 x y 2 8) Sketch the graph of y = 2 – 2 – 3. = y 1 + 2 = y = – 3 1 – 1 x (– 1) x 1 0 – 3– 4 – 3 0 1 (1) Review

27. For Educational Use Only © 2010 -3 -5 -3 1 3 5 1 3 5 -5 8) Sketch the graph of y = x 2 – 2x – 3. 0 – 3– 4 – 3 0 x y – 1 01 2 3 0 (-1,0) (0,-3) (2,-3) (3,0) (1,-4) 0 – 3 1 – 4 2 – 3 3 0 Review

28. For Educational Use Only © 2010 – 5 + 3 (x – 3) (x + 5) 9) Definition = 0 x – 3 x + 5 x – 5 = ( ) ( ) x + 3 = 3 All you need to graph are the x-intercepts & the vertex.

29. For Educational Use Only © 2010 (x + 5) (x – 3) 9) Definition All you need to graph are the x-intercepts & the vertex. = 0 y = x-intercepts = – 5 & 3 vertex = (Average of the x-int, #)

30. For Educational Use Only © 2010 – 16 (x + 5) (x – 3) 9) Definition = 0 y = x-intercepts = – 5 & 3 vertex = (Average of the x-int, #) – 5 + 3 2 = – 1, (? ) ( + 5) ( – 3) = – 16 y = – 1 3– 5 – 1 – 16 x x

31. For Educational Use Only © 2010 -4 -12 -20 -4 -12 4 12 20 4 12 20 -20 9) y = (x + 5)(x – 3) (-5,0) (3,0) (– 1,-16) Review x-intercepts vertex – 5 & 3 (– 1,– 16)

32. For Educational Use Only © 2010 – 1 1 – 2 + 6 (x – 6) (– x + 2) (x – 6) (– x + 2) 10) Definition = 0 x – 6 – x + 2( ) ( ) x + 6 = 6 All you need to graph are the x-intercepts & the vertex. – x – 2 = + 0 – 1 x=2

33. For Educational Use Only © 2010 (– x + 2) (x – 6) 10) Definition All you need to graph are the x-intercepts & the vertex. = 0 y = x-intercepts = 2 & 6 vertex = (Average of the x-int, #)

34. For Educational Use Only © 2010 4 = 4 4 44 (– x + 2) (x – 6) 10) Definition = 0 y = x-intercepts = 2 & 6 vertex = (Average of the x-int, #) 2 + 6 2 = 4, (? ) (– + 2) ( – 6) y = 62 4 4x x

35. For Educational Use Only © 2010 -2 -6 -10 -2 -6 2 6 10 2 6 -10 10) y = (– x + 2)(x – 6) (2,0) (6,0) (4,4) Review x-intercepts vertex 2 & 6 (4, 4)

36. For Educational Use Only © 2010 1) Don’t forget the negative signs. 2) You can sketch quadratic equations. Key Points & Don’t Forget

37. For Educational Use Only © 2010 pg. 407-408 #’s 11-18, 23-26, 31, 40, 41-49 odd The Assignment

38. For Educational Use Only © 2010 Please email brianspowerpoints@gmail.com with errors, confusing slides, improvements, complications, or any other comments or questions.brianspowerpoints@gmail.com Arch picture from http://flashydubai.com/670-feet-tall- worlds-tallest-arch-bridge-in-dubai/http://flashydubai.com/670-feet-tall- worlds-tallest-arch-bridge-in-dubai/ The template is from www.spiralgraphics.bizwww.spiralgraphics.biz http://www.worldofteaching.comhttp://www.worldofteaching.com is home to over a thousand powerpoints submitted by teachers. This is a completely free site and requires no registration. Please visit and I hope it will help in your teaching. Bibliography