anemone said:Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_0, a_a,\cdots,a_n$ are all smaller than 4 and are integer, $a_n \in (0, 1, 2,\cdots)$.
Given that $f(4)=2009$, find $f(1)$.
kaliprasad said:= 1 + 3 + 3 + 1 + 2 + 1 = 11
as f(x) = x^5 + 3x^4 + 3x^3 + x^2 +2x +1
as no coefficient is >4 and we are given f(4) subtract the highest power of 4 as many times as it can go
2009 = 1024 + 985
985 = 256 * 3 + 217
217 = 64 * 3 + 25
25 = 16 + 9
9 = 2 *4 + 1
$f(1)$ in a polynomial refers to the value of the polynomial when the variable is replaced with the number 1.
Integer coefficients in a polynomial are the numerical values that are multiplied by the variable terms. They can only be whole numbers (positive, negative, or zero).
The highest degree of a polynomial with integer coefficients $\leq$ 4 is 4. This means that the largest exponent in the polynomial will be 4.
To find $f(1)$ in a polynomial, simply replace every instance of the variable with the number 1 and then solve the resulting expression.
Finding $f(1)$ in a polynomial of integer coefficients $\leq$ 4 can help determine the value of the polynomial at a specific point, which can be useful in many mathematical applications. It can also provide information about the behavior and characteristics of the polynomial.