Zero-Product Property and Consecutive Numbers

Hi Class!

-On wednesday we went over polynomial equations and the zero-product property. A polynomial is an equation that is equivalent to an equation with a polynomial as one side of the = sign, and 0 on the other side. Examples of this type of equation are x^2 + 4x + 4 = 0, and x^4 + x= = 0. To solve these types of equations, you must factor the polynomial into linear factors. If you cannot factor the polynomial, then it is prime. 


-Explanation of the zero-product property in book: p183-185


-Visual example: △ x □  = 0

          ~Either △ = 0, □ = 0, or □ and △ = 0
          ~This can also be shown as: ab = 0
                                   where a = 0 and or b = 0
                 
           *There can be more than one value for X

-The steps to solving polynomial equations with the zero-product property are:
          
          1. Write the equation with one side as zero (set the equation equal to zero)
          2. Factor the non-zero side of the equation
          3. Set both of the factors equal to zero and solve

-Example problem:

          1. One side as zero:  x^2 - 49 = 0
          2. Factor:  (x+2) (x-2) = 0
          3. Set factors equal to zero:  x+2 = 0, x-2 = 0
                                                       x = 2, -2 ---> {2, -2}

-Alternate ways to solve: In class we also learned another way to solve certain problems:

          ~9x^2 - 81 = 0: Instead of factoring, you could also solve for x normally.
            9x^2 = 81
            x^2 = 9
            x = 3, -3

                                *Do not forget that the square root can be positive AND negative

          ~Here is the same problem (^), but using the zero-product property:
            9x^2 - 81 = 0
            (3x-9) (3x+9) = 0
            3x-9 = 0, 3x+9 = 0
            3x = 9, 3x = -9
            x = 3, x= -3
            {3, -3}


-Another example of zero-product property:

          ~x^2 + 10x + 25 = 0
            (x-5) (x-5) = 0 
            x = 5  
            ^When the factor occurs twice (same factor more than once) it's called a 'Double Root'

- Consecutive Numbers: In class we also learned how to express consecutive integers, or consecutive even integers and consecutive odd integers.

          ~For example: 2 consecutive odd integers numbers that multiply to 99
                                    x (x+2) = 99
                                           *x represents the first integer and x+2 represents the second integer
                                   x^2 + 2x = 99
                                   x^2 + 2x - 99 = 0
                                  (x+11) (x-9) = 0
                                   x = -11, x = 9 {-11, 9)
                                            *-11 and 9 are the two possibilities for x NOT the answers to the problem.
                                             To find the answer  ---> plug -11 and 9 into x+2 (add two)

         ~Another example: 2 consecutive even integers  ---> the difference of who's squares is 68
                                         x^2 - (x+2)^2 = 68
                                         (x+2)^2 -x^2 - 68 = 0
                                         x^2 + 4x + 4 - x^2  - 68 = 0
                                         4x - 64 = 0
                                         4x = 64
                                          x = 16     *Now plug into original equation
                                                           (16 + 2)^2 - 16^2 = 68
                                                            {16,18} {-16,-18} 



- http://www.youtube.com/watch?v=1Iay8rFBQ6o


                                                           Hope this helped!
                                           Problems are on pages 185-191 in the book!