For Educational Use Only © Solving Polynomial Equations in Factored Form Brian Preston Algebra

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

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

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

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

For Educational Use Only © 2010 – (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 =

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

For Educational Use Only © ) An arch is modeled by the equations y = – (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 = – (x – 315)(x + 315) Real World Application

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

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, #) – (x – 315)(x + 315) = 0 y = 3)

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

For Educational Use Only © 2010 x-intercepts 3) y = – (x – 315)(x + 315) (-315,0) (315,0) (0,630) Review vertex 315 & – 315 (0,630)

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

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

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

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

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( )

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

For Educational Use Only © 2010 – (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

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.

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

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 – – 1 – 4 x y (1) Review

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

For Educational Use Only © ) Sketch the graph of y = 2 – 2 – 3. = y 1 – 6 = y = – 3 9 – 9 x (3)(3) x (3)(3) 1 – 4 – – 1 x y (9) Review

For Educational Use Only © – 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 – – 1 (0) Review

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

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

For Educational Use Only © 2010 – (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.

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, #)

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

For Educational Use Only © ) y = (x + 5)(x – 3) (-5,0) (3,0) (– 1,-16) Review x-intercepts vertex – 5 & 3 (– 1,– 16)

For Educational Use Only © 2010 – 1 1 – (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

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, #)

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

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

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

For Educational Use Only © 2010 pg #’s 11-18, 23-26, 31, 40, odd The Assignment

For Educational Use Only © 2010 Please with errors, confusing slides, improvements, complications, or any other comments or Arch picture from worlds-tallest-arch-bridge-in-dubai/ worlds-tallest-arch-bridge-in-dubai/ The template is from 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