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110 3. Factors and Zeros

Show, by dividing w(t) by v(t), that there are polynomials ai (1 < i 5 Ic),
for which degai < degv and

w(t) = Ok(t) + ak-1(+(t) + ak-2(t)v(t)2 + ‘..+ al(t)v(t)k-l,

so that
w(t)=-+ ak @>
v(t)” v(t)”

ak-dt) +... + adt)

v(t)“-’ v(t)’

3.7 Problems on Factorization

1. Factor over 2:

(a) 4x4+ 1

(b) z4 - 20~~ + 4

(c) X2Y2 - (x + Y)XY + x + y - 1

(d) x2y2z2-(x+y+z)xy%+xy+y%+%x-1

(e) (b - c)~ + (c - a)” + (a - b)3

(f) a2(b + c) + b2(c + a) + abc - c3

(g) bc(b + c) + ca(c + a) -t ab(a + b) -I- 2abc

(h) bc(b + c) + ca(c - a) - ab(a + b)

(i) xl0 +x5+1

(j) 2x3 + 6xy2 + z3 - 3x2% + 3#2

(k) s~~+y~+%~+z~y~%~-2x~ y%-2xy2%-2xy%2+2xy+2x%+2y%-4

(1) x2(y3 - %3) + y2(z3 - x”) + %2(X” - ys)

(m) (a + b + c)~ - (b + c)~ - (c + a)” - (a + b)4 + a4 + b4 + c4

(n) (bc + ca + ab)3 - b3c3 - c3a3 - a3b3

(0) 2(bc + ca + ab)2 - a2(b + c)~ - b2(c + u)~ - c2(a + b)2

(P) 6(x5 + y” + z”) - 5(z2 + y2 + z”)(x” + y3 + z3)

(q) 2(x4 + y” + z4 + w4) - (x2 + y2 + z2 + w2)2 + 8xyzw

(r) x3y3 + y3r3 + %3z3 - x4y% - xy4% - xyz4

(s) (~2+y2+%2)(x+y+%)(2+y-%)(y+%-x)(%+x-y)-8x2y2%2

(t> x(y + g2 + Y(% + x)2 + %(X + y)2 - 4xyz

(u) a4 + b4 - c4 - 2a2b2 + 4abc2

2. Show that the two equations

x4 - x3 $ x2 + 2x - 6 = 0

x4 + x3 + 3x2 + 42 + 6 = 0

have a pair of complex roots in common.