3.7. Problems on Factorization 111
- Find a polynomial p(x) such that [P(x)]~ - x is divisible by (x - 1)
(x - 2)(x - 3). - Find values of a and b which will make
(ax + b)(x5 + 1) - (52 + 1)
divisible by z2 + 1. Check your answer.
- (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.
- 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. - 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. - For what integer a does x2 - z + a divide xl3 + x + 90?
- 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.
- 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. - Prove that 212 divides
3(81”+l) + (16n - 54)9”+’ - 320n2 - 1447~ + 243.
- 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.