Question

for the polynomial P(x)=4x^2-5x+6 and c =1, find p(c) by (a. direct substitution and (b. the remainder theorem

Answer 1

$p(x)=4x^2-5x+6$

$c=1\implies p(c)=4(1)^2-5(1)+6=5$

By the remainder theorem, the remainder upon dividing a polynomial $p(x)$ by a linear binomial $x-c$ is equal to $p(c)$. Via synthetic division, we get

1   |  4   -5   6
.    |        4  -1
- - - - - - - - - - -
.    |  4   -1   5

which translates to

$\dfrac{4x^2-5x+6}{x-1}=4x-1+\dfrac5{x-1}$

or

$4x^2-5x+6=(4x-1)(x-1)+5$

Indeed, when $x=1$, the first term on the right hand side vanished and we're left with 5.