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Let $Q(x) = ax^2+bx + c$ and put that into P. Plug in x =1,2,3 and now you have a system of three linear equations with three possibilities each. Now it's just combinatorics! Calculate the inverse matrix and it is easy to see that out of the 27, 5 do not work. This the answer is 22.

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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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## Comments

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TopNewestWhat are your thoughts? What have you tried?

Must the polynomial have complex, real or integer coefficients?

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What I have tried is :

Degree of P(x) is 3 and R(x) is also 3

So degree of P(x).R(x) is 6.

Degree of Q(x) must be 2 so that degree of P(Q(x)) is 6.

P(Q(x)) = P(x).R(x)

(Q(x)-1)(Q(x)-2)(Q(x)-3) = (x-1)(x-2)(x-3).R(x)

Now what I have to do ?

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Great job. A key piece of information is that the degree of Q(x) is 6.

What can we say about the value of $Q(1)$? Q(2)? Q(3)?

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(Q(x)-1)(Q(x)-2)(Q(x)-3) = (x-1)(x-2)(x-3).R(x)

[Q(1)-1][Q(1)-2][Q(1)-3] =0

Q(1)-1 = 0. So Q(1) =1

Q(1)-2 = 0. So Q(1) = 2

Q(1)-3 = 0. So Q(1) = 3

Hence Q(1) = 1or 2or 3

Similarly Q(2) = 1or 2or 3

And Q(3) = 1or 2or 3

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$Q(1), Q(2), Q(3)$.

Great, so we now know the possible values ofLets say that $Q(1) = 1, Q(2) = 2, Q(3) = 2$. How many polynomials $Q(x)$ of degree 2 satisfy that condition?

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lagrange interpolation

Check outHint:If we have $n$ equations of the form $f(x_i) = y _ i$, then there is a unique polynomial of degreeat most$n-1$ which satisfies the conditions.Log in to reply

I think the answer is 2

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Answer is 22

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Yup, that's the numerical answer. The more interesting part is the actual solution.

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Let $Q(x) = ax^2+bx + c$ and put that into P. Plug in x =1,2,3 and now you have a system of three linear equations with three possibilities each. Now it's just combinatorics! Calculate the inverse matrix and it is easy to see that out of the 27, 5 do not work. This the answer is 22.

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