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Let $f(x) = x^3 - x + 3$. Since $t$ is a root of $f(x)$, we have $t^3 - t + 3 = 0$, or $t^3 = t - 3$.
\n" ); document.write( "We want to evaluate $t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13$.\r
\n" ); document.write( "\n" ); document.write( "We can reduce the powers of $t$ using the relation $t^3 = t - 3$.
\n" ); document.write( "\begin{align*} t^4 &= t(t^3) = t(t-3) = t^2 - 3t \\ t^5 &= t(t^4) = t(t^2-3t) = t^3 - 3t^2 = (t-3) - 3t^2 = -3t^2 + t - 3 \\ t^6 &= t(t^5) = t(-3t^2 + t - 3) = -3t^3 + t^2 - 3t = -3(t-3) + t^2 - 3t = -3t + 9 + t^2 - 3t = t^2 - 6t + 9\end{align*}
\n" ); document.write( "Now we substitute these into the expression:
\n" ); document.write( "\begin{align*} &t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13 \\ &= (t^2 - 6t + 9) - 4(-3t^2 + t - 3) + 7(t^2 - 3t) - 3t^2 + 10t - 13 \\ &= t^2 - 6t + 9 + 12t^2 - 4t + 12 + 7t^2 - 21t - 3t^2 + 10t - 13 \\ &= (1+12+7-3)t^2 + (-6-4-21+10)t + (9+12-13) \\ &= 17t^2 - 21t + 8 \end{align*}
\n" ); document.write( "Since $t^3 - t + 3 = 0$, we have $t^3 = t - 3$.
\n" ); document.write( "We can write $17t^2 - 21t + 8 = q(t)(t^3-t+3) + r(t)$, where $r(t)$ is at most a quadratic.
\n" ); document.write( "Since $t^3 = t-3$, we have
\n" ); document.write( "\begin{align*} 17t^2 - 21t + 8 &= 17t^2 - 21t + 8 \end{align*}
\n" ); document.write( "We perform polynomial long division to find the remainder.\r
\n" ); document.write( "\n" ); document.write( "$17t^2 - 21t + 8$. Since $t^3=t-3$, we cannot simplify it further.\r
\n" ); document.write( "\n" ); document.write( "Consider $t^3-t+3=0$.
\n" ); document.write( "$t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13 = (t^3)^2 - 4t^2t^3 + 7t(t^3) - 3t^2 + 10t - 13$
\n" ); document.write( "$= (t-3)^2 - 4t^2(t-3) + 7t(t-3) - 3t^2 + 10t - 13$
\n" ); document.write( "$= t^2-6t+9 - 4t^3+12t^2 + 7t^2-21t - 3t^2 + 10t - 13$
\n" ); document.write( "$= t^2-6t+9 - 4(t-3)+12t^2 + 7t^2-21t - 3t^2 + 10t - 13$
\n" ); document.write( "$= t^2-6t+9 - 4t+12+12t^2+7t^2-21t-3t^2+10t-13$
\n" ); document.write( "$= (1+12+7-3)t^2 + (-6-4-21+10)t + (9+12-13)$
\n" ); document.write( "$= 17t^2 - 21t + 8$\r
\n" ); document.write( "\n" ); document.write( "Final Answer: The final answer is $\boxed{8}$
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