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3.7. Problems on Factorization 111

3. Find a polynomial p(x) such that [P(x)]~ - x is divisible by (x - 1)
   (x - 2)(x - 3).


4. Find values of a and b which will make

(ax + b)(x5 + 1) - (52 + 1)

divisible by z2 + 1. Check your answer.

5. (a) If ax3 + bx + c, where a # 0, c # 0, has a factor of the form

x2+px+1,

show that

a2- c2 = ab ,

(b) In this case, prove that ax3 + bx + c and cc3 + bx2 + a have a

common quadratic factor.

6. If the quadratic function 3x2 + Ppxy + 2y2 + 2ax - 4y + 1 can be
   resolved into factors linear in x and y, prove that p must be a root of
   the equation
   p2+4ap+2a2+6=0.


7. A manic cubic polynomial over Z has the property that one of its
   zeros is the product of the other two. Show that it must be reducible
   over Z.


8. For what integer a does x2 - z + a divide xl3 + x + 90?


9. Show that

b2(a - b)(c - b){(a - b)2 + (c - b)2} - ab2c(a2 + c2) + b5(a - b-i-c)

is the cube of a polynomial.

10. Determine all values of the parameters a and b for which the polyno-
    mial
    x4 + (20 + 1)x3 + (a - 1)2x2 + bx + 4
    can be factored into a product of two manic quadratic polynomials
    p(x) and q(x) such that the equation q(x) = 0 has two different roots
    r and s with p(r) = s and p(s) = r.


11. Prove that 212 divides

3(81”+l) + (16n - 54)9”+’ - 320n2 - 1447~ + 243.

12. Let j(x) = xn + xa + 1 be a polynomial over Z2 such that 0 < a < n.
    Show that, if f(x) h as any repeated factors, then f(x) is a perfect
    square.