F.2 Maths Factorization

1. m^ 4- 18m^ 2+ 81
[m^2]^2 - 18(m^ 2) + 81
(m^2 - 9)((m^2 - 9)
(m +3)(m - 3)(m +3)(m - 3)
(m +3)^2 (m - 3)^2

2. 16p^ 4- 8p^ 2q^2+ q^4
16(p^ 2)^2 - 8(pq)^2+ (q^2)^2
(4p^2 - q^2)(4p^2 - q^2)
((2p)^2 - q^2)((2p)^2 - q^2)
(2p +q)(2p - q)(2p +q)(2p - q)
(2p +q)^2 (2p - q)^2

m^4 = (m^2)^2

m^ 4- 18m^ 2+ 81
= (m^2)^2 - 2．9．m^2 + 9^2
= (m^2 - 9)^2

or let A = m^2

then
m^ 4- 18m^ 2+ 81
= A^2 - 2．9．A + 9^2
= (A - 9)^2
= (m^2 - 9)^2

Similar
16p^ 4- 8p^2q^2+ q^4
= (4p^ 2)^2 - 2．4p^ 2．q^2 + (q^2)^2
= (4p^ 2 - q^2)^2

2013-09-06 13:22:31 補充：
a^2 - b^2 = (a+b)(a-b)

m^2 - 9 = (m^2 - 3^2) = (m+3)(m-3)

=>
m^ 4- 18m^ 2+ 81
= (m^2 - 9)^2
= [(m+3)(m-3)]^2
= (m+3)^2(m-3)^2


4p^ 2 - q^2 = (2p)^2 - q^2 = (2p + q)(2p - q)

=>
16p^ 4- 8p^2q^2+ q^4
= (2p + q)^2．(2p - q)^2

2013-09-07 02:09:18 補充：
https://www.dropbox.com/s/1bpt9fyqj2kp5sn/20130906PIC02.png