mAtHs 唔該幫幫手

use the following identities for factorization:
(a+b)^2 = a^2 + 2ab + b^2
(a-b)^2 = a^2 + 2ab + b^2
a^2 - b^2 = (a+b)(a-b)
(a+b)^3 = (a+b)(a^2 - ab + b^2)
(a -b)^3 = (a-b)(a^2 + ab + b^2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1. c^4 - 1
= (c^2 -1)(c^2 +1)
= (c+1)(c-1)(c^2+1)

2. 54x^3+250y^3
= 2 (27x^3 + 125y^3)
= 2 [(3x)^3 + (5y)^3]
= 2 (3x+5y) [(3x)^2 + (3x)(5y) + (5y)^2]
= 2 (3x+5y) (9x^2 + 15xy + 25y^2)

3. (m+1)^3 - (m-1)^3
= [(m+1) - (m-1)] [ (m+1)^2 + (m+1)(m-1) + (m-1)^2]
= 2 [ (m^2 + 2m + 1) + (m^2 - 1) + (m^2 - 2m + 1) ]
= 2 (3m^2 + 1)

4. 500 - 32m^3n^3
= 4 (125 - 8m^3n^3)
= 4 [ 5^3 - (2mn)^3 ]
= 4 (5 - 2mn) (5^2 + 10mn + 4m^2n^2)

1)c^4-1
=(c²)²-1²
=(c²+1)(c²-1)
=(c²+1)(c+1)(c-1)


2)54x³+250y³
=2(27x³+125y³)
=2[(3x)³+(5y)³]
=2(3x+5y)(9x²-15xy+25y²)


3)(m+1)³-(m-1)³
=[(m+1)-(m-1)][(m+1)²+(m+1)(m-1)+(m-1)²]
=2(m²+2m+1+m²-1+m²-2m+1)
=2(3m²+1)

4)500-32m³n³
=4(125-8m³n³)
=4[5³-(2mn)³]
=4(5-2mn)(25+10mn+4m²n²)

1 c^4 -1
= (c^2)^2 -1^2
= (c^2-1)(c^2+1)
= (c-1)(c+1)(c^2+1)


2 54x^3+250y^3
= 2(27x^3+125y^3)
= 2[ (3x)^3+(5y)^3]
= 2[ (3x+5y)( (3x)^2-(3x)(5y)+(5y)^2 ) ]
= 2[ (3x+5y)(9x^2-15xy+25y^2) ]


3 (m+1)^3-(m-1)^3
= [(m+1)-(m-1)][(m+1)^2+(m+1)(m-1)+(m-1)^2]
= 2 [(m+1)^2 + m^2 -1 + (m-1)^2]
= 2 [m^2 + 2m +1 + m^2 -1 + m^2 - 2m +1 ]
= 2 [3m^2 - 1 ]


4 500-32m^3n^3
= 4 {125-8m^3n^3}
= 4 {(5)^3 - (2mn)^3]
= 4{ (5-2mn)((5)^2 + 5(2mn) +(2mn)^2)}
= 4 { (5-2mn)(25 + 10mn +4m^2n^2) }