a)
$x^{4}$ +$x^{2}$ +1
=$x^{4}$ -x+$x^{2}$+x +1
=x($x^{3}$-1) +$x^{2}$+x +1
=x(x-1)($x^{2}$+x +1) +$x^{2}$+x +1
=($x^{2}$-x)($x^{2}$+x +1) +$x^{2}$+x +1
= ($x^{2}$+x +1)($x^{2}$-x +1)
b)
$x^{3}$-19x+30
= $x^{3}$ -25x+6x+30
=x($x^{2}$ -25) +6(x+5)
=x(x+5)(x-5)+6(x+5)
=(x+5)($x^{2}$ +5x+6)
=(x+5)($x^{2}$+2x+3x+6)
=(x+5)[x(x+2)+3(x+2)]
=(x+5)(x+2)(x+3)
c)
ab(a-b)+bc(b-c)+ca(c-a)
=a2b - ab2 + b2c - bc2 + a2c - ac2
= abc +a2b - ab2 +b2c - bc2 +a2c - ac2 - abc
= (a2b - abc) - (ab2 - b2c) - (bc2 - ac2) - (a2c - abc)
= ab(a - c) - b2(a - c) - c2(b - a) - ac(a - b)
= [ab(a - c) - b2(a - c)] + [c2(a - b) - ac(a - b)]
= (a - c)(ab - b2) + (a - b)(c2 - ac)
= b(a - c)(a - b) + c(a - b)(c - a)
= b(a - c)(a - b) - c(a - b)(a - c)
= (a - c)(a - b)(b - c)
