2005 AMC 12A Problems/Problem 24

Problem

Let $P(x)=(x-1)(x-2)(x-3)$ . For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree 3 such that $P(Q(x))=P(x) \cdot R(x)$ ?

$\mathrm {(A) } 19 \qquad \mathrm {(B) } 22 \qquad \mathrm {(C) } 24 \qquad \mathrm {(D) } 27 \qquad \mathrm {(E) } 32$

Solution

We can write the problem as

$P(Q(x))=(Q(x)-1)(Q(x)-2)(Q(x)-3)=P(x)\cdot R(x)=(x-1)(x-2)(x-3)\cdot R(x)$ .

Since $\deg P(x) = 3$ and $\deg R(x) = 3$ , $\deg P(x)\cdot R(x) = 6$ . Thus, $\deg P(Q(x)) = 6$ , so $\deg Q(x) = 2$ .

$P(Q(1))=(Q(1)-1)(Q(1)-2)(Q(1)-3)=P(1)\cdot R(1)=0,$
$P(Q(2))=(Q(2)-1)(Q(2)-2)(Q(2)-3)=P(2)\cdot R(2)=0,$
$P(Q(3))=(Q(3)-1)(Q(3)-2)(Q(3)-3)=P(3)\cdot R(3)=0.$

Hence, we conclude $Q(1)$ , $Q(2)$ , and $Q(3)$ must each be $1$ , $2$ , or $3$ . Since a quadratic is uniquely determined by three points, there can be $3*3*3 = 27$ different quadratics $Q(x)$ after each of the values of $Q(1)$ , $Q(2)$ , and $Q(3)$ are chosen.

However, we have included $Q(x)$ which are not quadratics: lines. Namely,

$Q(1)=Q(2)=Q(3)=1 \Rightarrow Q(x)=1,$
$Q(1)=Q(2)=Q(3)=2 \Rightarrow Q(x)=2,$
$Q(1)=Q(2)=Q(3)=3 \Rightarrow Q(x)=3,$
$Q(1)=1, Q(2)=2, Q(3)=3 \Rightarrow Q(x)=x,$
$Q(1)=3, Q(2)=2, Q(3)=1 \Rightarrow Q(x)=4-x.$

Clearly, we could not have included any other constant functions. For any linear function, we have $2\cdot Q(2) = Q(1) + Q(3)$ because $Q(2)$ is y-value of the midpoint of $(1, Q(1))$ and $(3, Q(3))$ . So we have not included any other linear functions. Therefore, the desired answer is $27 - 5 = \boxed{\textbf{(B) }22}$ .

Quicker Solution: We see that

$P(Q(x))=(Q(x)-1)(Q(x)-2)(Q(x)-3)=P(x)\cdot R(x)=(x-1)(x-2)(x-3)\cdot R(x)$ .

Therefore, $P(x) | P(Q(x))$ . Since $deg Q = 2,$ we must have $x-1, x-2, x-3$ divide $P(Q(x))$ so we pair them off with one of $Q(x)-1, Q(x)-2, Q(x)-3$ to see that there are $3!+3 \cdot 2 \cdot \binom{3}{2}$ = 24 without restrictions (the count was made by pairing off the linear factors of $P(x)$ with $Q(x)-1, Q(x)-2, Q(x)-3$ and notice that the degree of $Q$ is 2). However, we have two functions which are constant, which are $Q(x) = x$ and $Q(x) = 4-x.$ The answer is $22$ .

~Williamgolly

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2005 AMC 12A Problems/Problem 24

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