Can someone help please?? These are 4 questions

Question

You are saving $25 a week to buy a new HDTV. Your sister is saving at a weekly rate of 5x-7, if x = the number of weeks of saving. In how many weeks will you both have the same amount of money?

 

Use the Factor Theorem to find the solutions to x3+2x2 - 104x + 192 = 0 if x - 2 is a factor. Which is a solution?

 

Given h(x) = x-3 and f(x) = x3-x2-x-15. Is h(x) a factor of f(x)?

 

Given h(x) = x+2 and f(x) = x3 +4x2-5. Is h(x) a factor of f(x)?

 

 

Answer 1

 

25x = x(5x - 7)

 

25x = 5x^2 - 7x

 

0 = 5x^2 - 32x

 

0 = x(5x-32)

  

x=0 weeks gets ignored.

 

 x = 32/5 weeks

 

The sister saves

5(32/5) - 7 = 32-7 = 25 for that week.

 

25*32/5 = 5 * 32 = 160

 

vs.

 

25*32/5 = 5*32 = 160

 

They will have the same amount after 32/5 = 6.4 weeks or 6 months and 12 days

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x-2 is a factor ---> x = 2 is a solution

Synthetic division:

   

  2 |    1    2     -104     192

                2         8     -192

      -----------------------------

          1    4       -96       0

 

x^2 + 4x - 96 = 0

(x   +  12   )(x    -   8  ) = 0

 

x+12=0 ---> x = -12 

 or

x - 8 = 0 ---> x = 8

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f(3) = 3^3 - 3^2 - 3 - 15 = 27 - 9 - 3 - 15 = 18 - 3 - 15 = 15 - 15 = 0

So 3 is a solution.

 

 3 | 1   -1     -1     -15

            3      6      15

     __________________

      1    2      5      0

 

x^2 + 2x + 5 = 0

x = (-2 +or- i)/2

 

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f(-2) = (-2)^3 + 4(-2)^2 - 5 =

           -8 + 4*4 - 5

            -8 + 16 - 5

             8 - 5 = 3

 No -2 is not a solution