Difference between revisions of "2014 AMC 12B Problems/Problem 16"

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<cmath>P(2)+P(-2)=12k+2k=\boxed{\textbf{(E)}\ 14k}</cmath>
 
<cmath>P(2)+P(-2)=12k+2k=\boxed{\textbf{(E)}\ 14k}</cmath>
 
==Solution 2==
 
==Solution 2==
If we use Gregory's Triangle, the following happens.
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If we use [[Gregory's Triangle]], the following happens:
 
<cmath>P(-1), P(0), P(1)</cmath>
 
<cmath>P(-1), P(0), P(1)</cmath>
 
<cmath>  3k ,  k  ,  2k </cmath>
 
<cmath>  3k ,  k  ,  2k </cmath>
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The above shows us that <math>P(-2)</math> is <math>8k</math> and <math>P(2)</math> is <math>6k</math> so <math>8k+6k=14k</math>.
 
The above shows us that <math>P(-2)</math> is <math>8k</math> and <math>P(2)</math> is <math>6k</math> so <math>8k+6k=14k</math>.
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NOTE (not from author): The link you put for gregory's triangle doesn't work so please explain it in your post or find a resource that does work; there isn't much on google.
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2014|ab=B|num-b=15|num-a=17}}
 
{{AMC12 box|year=2014|ab=B|num-b=15|num-a=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:34, 1 November 2024

Problem

Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ k\qquad\textbf{(C)}\ 6k\qquad\textbf{(D)}\ 7k\qquad\textbf{(E)}\ 14k$

Solution

Let $P(x) = Ax^3+Bx^2 + Cx+D$. Plugging in $0$ for $x$, we find $D=k$, and plugging in $1$ and $-1$ for $x$, we obtain the following equations: \[A+B+C+k=2k\] \[-A+B-C+k=3k\] Adding these two equations together, we get \[2B=3k\] If we plug in $2$ and $-2$ in for $x$, we find that \[P(2)+P(-2) = 8A+4B+2C+k+(-8A+4B-2C+k)=8B+2k\] Multiplying the third equation by $4$ and adding $2k$ gives us our desired result, so \[P(2)+P(-2)=12k+2k=\boxed{\textbf{(E)}\ 14k}\]

Solution 2

If we use Gregory's Triangle, the following happens: \[P(-1), P(0), P(1)\] \[3k ,  k  ,  2k\] \[-2k , k\] \[3k\]

Since this is cubic, the common difference is $3k$ for the linear level so the string of $3k$s are infinite in each direction. If we put a $3k$ on each side of the original $3k$, we can solve for $P(-2)$ and $P(2)$.

\[P(-2), P(-1), P(0), P(1), P(2)\] \[8k ,   3k ,  k  ,  2k ,  6k\] \[-5k  , -2k ,  k  ,  4k\] \[3k   ,  3k ,  3k\]

The above shows us that $P(-2)$ is $8k$ and $P(2)$ is $6k$ so $8k+6k=14k$.

NOTE (not from author): The link you put for gregory's triangle doesn't work so please explain it in your post or find a resource that does work; there isn't much on google.

See also

2014 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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All AMC 12 Problems and Solutions

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