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1.9. Other Problems 43

If y is nonzero, deduce that

Y + (b2 - Wq + q2)/y + 2(ap - q) = o.

7. Find the square roots of 1 - x + d22x - 15 - 8x2.


8. Determine necessary and sufficient conditions that ax4 + bx3 + cx2 +
   dx+e (u # 0) is of the formp(q(x)), where p and q are both quadratic.


9. From the pair of equations

x= l-v+(v/u), y= l-u+(u/v),

deduce the pair of equations

u=l-y+(y/x), v=l-x+(x/y),

and conversely.

10. Solve the equation

(x - 2)(x - 3)(x - 4)(x - 5) = 360.

11. Show that the polynomial x4y2+y4~2+~4x2-3x2y2~2 always assumes
    a nonnegative value when I, y, z are real, but cannot be written as
    the sum of squares of polynomials over R in x, y, Z.


12. Express x4+y4+x2+y2 as the sum of the squares of three polynomials
    over R in x, y.


13. Let P(x,y) = x2y+xy2 and &(x,y) = x2+xy+y2. For each positive
    integer n, define

Fn(x, y) = (x + y)” - xn - y”

Gn(x, y) = (x + y)” + x” + y”.

Observe that Gs = 2Q, F3 = 3P, G4 = 2Q2, Fs = 5PQ, Gg =

2Q3 + 3P2. Prove that, for each positive integer n, either F,., or G,

is expressible as a polynomial in P and Q over Z.

14. Define a sequence of polynomials P,(x, y, z) as follows:

Po(x, Y, %) = 1

Pm(X,Y,Z) = (x + %>(Y + z)Pm-l(X,Y,Z + 1) - ~2L-l(~,Y,4

Prove that each P,(x, y, Z) is symmetric in x, y, Z.