Nonnegative Polynomial Evaluation
Let $ Q $ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\]where $ a_0,\ldots,a_n $ are nonnegative integers. Given that $ Q(1)=4 $ and $ Q(5)=152 $, find $ Q(6) $.
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- $\frac{a}{b}$
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- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
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- $\pi$
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Solution
If $ n \ge 4, $ then $ Q(5) \ge 5^4 = 625, $ so $ n \le 3, $ and we can write
\[Q(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0.\]We have that $ Q(1) = a_3 + a_2 + a_1 + a_0 = 4, $ so $ a_i \le 4 $ for all $ i $. Also,
\[Q(5) = 125a_3 + 25a_2 + 5a_1 + a_0 = 152.\]Clearly, $ a_3 \le 1 $. If $ a_3 = 0, $ then $ 25a_2 + 5a_1 + a_0 = 152 $. But $ 25a_2 + 5a_1 + a_0 \le 25 \cdot 4 + 5 \cdot 4 + 4 = 125, $ so $ a_3 = 1 $.
Then
\[25a_2 + 5a_1 + a_0 = 27.\]Clearly, $ a_2 \le 1 $. If $ a_2 = 0, $ then $ 5a_1 + a_0 = 27 $. But $ 5a_1 + a_0 \le 5 \cdot 4 + 4 = 24, $ so $ a_2 = 1 $.
Then
\[5a_1 + a_0 = 2.\]It follows that $ a_1 = 0 $ and $ a_0 = 2, $ so
\[Q(x) = x^3 + x^2 + 2.\]In particular, $ Q(6) = 6^3 + 6^2 + 2 = \boxed{254} $.
Freeflows