![\"\"](\"/images/stars/stars-12.gif\")
You can put this solution on YOUR website!
Let $p(x)$ be a cubic polynomial. We are given that $p(0) = 0$, $p(1) = 1$, $p(2) = 2$, and $p(3) = 3$.\r
\n" ); document.write( "\n" ); document.write( "Consider the polynomial $q(x) = p(x) - x$. Then $q(0) = p(0) - 0 = 0 - 0 = 0$, $q(1) = p(1) - 1 = 1 - 1 = 0$, $q(2) = p(2) - 2 = 2 - 2 = 0$, and $q(3) = p(3) - 3 = 3 - 3 = 0$.
\n" ); document.write( "Since $q(x)$ has roots at $x = 0, 1, 2, 3$, and $p(x)$ is a cubic polynomial, $q(x)$ must be a quartic polynomial.
\n" ); document.write( "However, since $p(x)$ is a cubic polynomial, $p(x) - x$ can be at most a cubic polynomial. Thus, we must have $q(x) = ax(x-1)(x-2)(x-3)$ for some constant $a$.
\n" ); document.write( "Then $p(x) = q(x) + x = ax(x-1)(x-2)(x-3) + x$.\r
\n" ); document.write( "\n" ); document.write( "Since $p(x)$ is a cubic polynomial, the coefficient of $x^4$ must be zero. The $x^4$ term in $ax(x-1)(x-2)(x-3)$ is $ax^4$. Since $p(x)$ is a cubic, $a$ must be 0.
\n" ); document.write( "Therefore, $q(x) = 0$ and $p(x) = x$.\r
\n" ); document.write( "\n" ); document.write( "Then $p(4) = 4$.\r
\n" ); document.write( "\n" ); document.write( "However, if $p(x)$ is a cubic, then $q(x)$ is also a cubic.
\n" ); document.write( "Since $q(0)=q(1)=q(2)=q(3)=0$, $q(x)$ must be identically 0, which means $p(x)=x$.
\n" ); document.write( "Then $p(4) = 4$.\r
\n" ); document.write( "\n" ); document.write( "Consider the finite differences.
\n" ); document.write( "x | p(x) | Δp | Δ²p | Δ³p
\n" ); document.write( "---|---|---|---|---
\n" ); document.write( "0 | 0 | 1 | 0 | 0
\n" ); document.write( "1 | 1 | 1 | 0 |
\n" ); document.write( "2 | 2 | 1 | |
\n" ); document.write( "3 | 3 | | |\r
\n" ); document.write( "\n" ); document.write( "Since the third differences are zero, the polynomial is at most quadratic. Since the first differences are all 1, the polynomial is $p(x) = x$. Then $p(4) = 4$.\r
\n" ); document.write( "\n" ); document.write( "Final Answer: The final answer is $\boxed{4}$
\n" ); document.write( " \n" ); document.write( "