How to do this remainder theorem question?

Question

Suppose h(t) = -5t^2+15t+1 represents the approximate height in metres of a javelin t seconds after it is thrown.
Write a statement that corresponds to the quoteint h(t)/(t-b) where b is a positive integer.
Answer is -5t^2+15t+1=(t-b)(-5t-5b+15)-5b^2+15b+1.
How do i get this?
Show that the statement in part a may be written as Q(t)=(h(t)-h(b))/(t-b).
Answer is ((-5t^2+15t+1)-(-5b^2+15b+1))/(t-b)
Again, how do i get this?

Answer 1

Use synthetic division to divide -5t²+15t+1 by t-b:

 

 

b √-5          15              1

 

                 -5b          15b-5b²

 

     -5      -5b+15    -5b²+15b+1

 

The divisor is (t-b), the quotient is -5t-5b+15, and the remainder is -5b²+15b+1.

Answer 2

Use synthetic ddivision.

 

Since you are dividing h(t) by t - b, divide by b thusly:

 

  b | -5   15           1

     |____     -5b__     -5b²

       -5   15 - 5b     1 - 5b²

So h(t)/(t - b) = Q(t) + R(t)/(t - b)  Or Q(t) = h(t)/(t - b) - R(t)/(t - b).

 

This means that (-5t² + 15t + 1)/(t - b) = -5t + (15 - 5b) + (1 - 5b²)/(t - b)

 

Multiplying both sides by (t-b), this becomes:  -5t² + 15t + 1 = (t - b)(-5t + 15 - 5b) + 1 - 5b²