Question 935882
answer to question 1:


ax^3 + bx^2 + cx + d = (x-r) * (  ax^2 + (ra+b)x + (r^2a+rb+c)  )


answer to question 2:


ar^3 + br^2 + cr + d must be equal to 0


this is because r is a root of the equation which means that the value of f(r) must be equal to 0.


answer to question 3:


value of f(r) must be equal to 0.


this is because f(r) is equal to ar^3 + br^2 + cr + d


when r is a root, f(r) will be equal to 0.


r is a root when the remainder of the division by (x-r) is equal to 0.


when r is not a root, f(r) will be equal to the remainder of the division.


the synthetic division is shown below:


the first part shows the steps in the synthetic division.
the second part shows the resulting equation after the division has been performed.
the third part shows that the remainder must be equal to 0 if r is a root of the equation which occurs if (x-r) is a factor of the equation.


<img src = "http://theo.x10hosting.com/2015/010202.jpg" alt = "$$$" </>