The remainder theorem says that the remainder upon dividing a polynomial
![p(x)](https://tex.z-dn.net/?f=p%28x%29)
by a linear polynomial
![x-a](https://tex.z-dn.net/?f=x-a)
is the same as the value of
![p(x)](https://tex.z-dn.net/?f=p%28x%29)
at
![x=a](https://tex.z-dn.net/?f=x%3Da)
. Dividing by any linear polynomial will always result in the following:
![p(x)=(x-a)q(x)+r(x)](https://tex.z-dn.net/?f=p%28x%29%3D%28x-a%29q%28x%29%2Br%28x%29)
where
![q(x)](https://tex.z-dn.net/?f=q%28x%29)
and
![r(x)](https://tex.z-dn.net/?f=r%28x%29)
are also polynomials. Taking
![x=a](https://tex.z-dn.net/?f=x%3Da)
, the term involving
![q(x)](https://tex.z-dn.net/?f=q%28x%29)
vanishes, so that
![p(a)=r(a)](https://tex.z-dn.net/?f=p%28a%29%3Dr%28a%29)
is exactly the remainder upon dividing.
Via synthetic division, we have
... | 2 -9 7 -5 11
4 | 8 -4 12 28
- - - - - - - - - - - - - - - - - -
... | 2 -1 3 7 39
which translates to
![\dfrac{2x^4-9x^3+7x^2-5x+11}{x-4}=2x^3-x^2+3x+7+\dfrac{39}{x-4}](https://tex.z-dn.net/?f=%5Cdfrac%7B2x%5E4-9x%5E3%2B7x%5E2-5x%2B11%7D%7Bx-4%7D%3D2x%5E3-x%5E2%2B3x%2B7%2B%5Cdfrac%7B39%7D%7Bx-4%7D)
that is, we're left with a remainder of 39.
Via the remainder theorem, we have
![p(4)=2\times4^4-9\times4^3+7\times4^2-5\times4+11=39](https://tex.z-dn.net/?f=p%284%29%3D2%5Ctimes4%5E4-9%5Ctimes4%5E3%2B7%5Ctimes4%5E2-5%5Ctimes4%2B11%3D39)
as expected.
The train travels at 5.5km per hour.
We know that in x hours it will advance by x*5.5 km.
Hence we need to solve x*5.5=82.5, hence x=82.5/5.5=15
It will take 5 hours
Whatever you multiplied to the denominator, do it to the numerator.