Amber Quinn

2022-06-27

Let $f\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}....+{a}_{n}{x}^{n}$, where ${a}_{i}$ are non-negative integers for $i=0,1,2,....,n$. If $f\left(1\right)=21$ and $f\left(25\right)=78357$, find the value of $\frac{f\left(10\right)+3}{100}$

America Barrera

Expert

Following the hint in the comment, noting that ${25}^{3}<78357<{25}^{4}$ shows that $n\le 3$, i.e. $f\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+{a}_{3}{x}^{3}$
From here, just do long division to find ${a}_{3},{a}_{2},{a}_{1},{a}_{0}$. For instance, $78357={25}^{3}\cdot 5+232$
, and the remainder 232 is $232={25}^{2}\cdot 0+25\cdot 9+7$

Do you have a similar question?