Question

Descompuneti in factori : a) (3x+1)²-(x+3)² b) (3x+5)²-(x+1)² c) (x+√2)²-8 d) (2x+√3)²-27 e) (√2x-1)²-32x² f) a⁶-b⁶ g ) x²-2√3x+3-(y-2√3)²

Answer 1

a) (3x+1)²-(x+3)²  =
 =(3x+1+x+3)(3x+1-x-3)  =
 =(4x+4)(2x-2)  =
=8(x+1)(x-1) 

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

c) (x+√2)²-8=
= (x+√2)²-(2√2)²=
= (x+√2+2√2)(x+√2-2√2)=
= (x+3√2)(x-√2)

d) (2x+√3)²-27 =
= (2x+√3)²-(3√3)² =
= (2x+√3)²-(3√3)² =
=(2x+√3+3√3)(2x+√3-3√3) =
=(2x+4√3)(2x-2√3) =
=4(x+2√3)(x-√3) 

e) (√2x-1)²-32x²  =
=(√2x-1)²-(4√2x)² =
=(√2x-1+4√2x)(√2x-1-4√2x) =
=(5√2x-1)(-3√2x-1)

f) a⁶-b⁶  =
= (a³-b³)(a³+b³)=
= (a-b)(a²+ab+b²)(a+b)(a²-ab+b²)

g ) x²-2√3x+3-(y-2√3)² =
= (x-√3)²-(y-2√3)² =
= (x-√3+y-2√3)(x-√3-y+2√3) =
= (x+y-3√3)(x-y+√3)