maths

1a)
x^4 - 6x^2 + 9
= (x^2)^2 - 6x^2 + 3^2
= (x^2 - 3)^2

1b)
x^4 - 6x^2 + 9 - 4x^2
= (x^2 - 3)^2 - 4x^2
= (x^2 - 3)^2 - (2x)^2
= (x^2 - 3 + 2x)(x^2 - 3 - 2x)
= (x^2 + 2x - 3)(x^2 - 2x - 3)

2)
x^6 - 1
= (x - 1)(x^5 + x^4 + x^3 + x^2 + x + 1)

3a)
x^4 + x^2 + 1
= x^4 + 2x^2 + 1 - x^2
= (x^2)^2 + 2x^2 + 1 - x^2
= (x^2 + 1)^2 - x^2
= (x^2 + 1 + x)(x^2 + 1 - x)
= (x^2 + x + 1)(x^2 - x + 1)

3b)
Put y+1 into x
x^4 + x^2 + 1
= (y+1)^4 + (y+1)^2 + 1
= [(y+1)^2 + (y+1) + 1] [(y+1)^2 - (y+1) +1] by using (a)
= (y^2 + 2y + 1 + y + 1 + 1)(y^2 + 2y + 1 - y - 1 +1)
= (y^2 + 3y + 3)(y^2 + y + 1)

Sub y = x
Therefore,
(x+1)^4 + (x+1)^2 + 1 = (x^2 + 3x + 3)(x^2 + x + 1)

4)
four different factors of degree 1
That is 4 different roots
x(x - 1)(x - 2)(x - 3)
= x^4 - 6x^3 + 11x^2 - 6x

two different factors of degree 2
x^2(x^2-1) = x^4 - x^2

one factors of degree 2 and two factors of degree 1
x^2(x-1)(x-2) = x^4 - 3x^3 + 2x^2