# Find nonconstant polynomials P(x), Q(x), R(x), and S(x) with integer coefficients such that `[[P(x),Q(x)],[R(x),S(x)]]^2` = `[[1,0],[0,1]]`.

One set of polynomials that satisfy the given conditions is P = 1 + x, Q = 2 + x, R = -x, and S = -1 - x. P(x), Q(x), R(x), and S(x) are nonconstant polynomials with integer coefficients. Some such polynomials need to be found such that ` [[P(x),Q(x)],[R(x),S(x)]]^2 = [[1,0],[0,1]] `.

First, square the matrix `[[P(x),Q(x)],[R(x),S(x)]]`

`[[P,Q],[R,S]]*[[P,Q],[R,S]] `

=`[[P^2+QR,PQ+QS],[PR+RS,QR+S^2]]`

`[[P^2+QR,PQ+QS],[PR+RS,QR+S^2]] = [[1,0],[0,1]]`

=> P^2+QR = 1 , PQ+QS = 0, PR+RS = 0, QR+S^2 = 1

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P(x), Q(x), R(x), and S(x) are nonconstant polynomials with integer coefficients. Some such polynomials need to be found such that ` [[P(x),Q(x)],[R(x),S(x)]]^2 = [[1,0],[0,1]] `.

First, square the matrix `[[P(x),Q(x)],[R(x),S(x)]]`

`[[P,Q],[R,S]]*[[P,Q],[R,S]] `

=`[[P^2+QR,PQ+QS],[PR+RS,QR+S^2]]`

`[[P^2+QR,PQ+QS],[PR+RS,QR+S^2]] = [[1,0],[0,1]]`

=> P^2+QR = 1 , PQ+QS = 0, PR+RS = 0, QR+S^2 = 1

PQ+QS = 0 => P+S = 0.

There are many polynomials P, Q, R, and S that satisfy the equations we have obtained.

If we take P = 1+x, P^2 = 1+2x+x^2, QR = -2x-x^2. Let Q = 2+x, R = -x, S = -P -(1+x).

This gives the polynomials P = 1+x, Q = 2+x, R = -x, and S = -1-x.

One set of polynomials that satisfy the given conditions is P = 1+x, Q = 2+x, R = -x ,and S = -1-x.

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