Question

CMR

a) (a²-b²)²+(2ab)²= (a²+b²)²

b) (ax+b)²+(a-bx)²+c²x²+c²=(a²+b²+c²).(x²+1)

c) (a+b+c)³= a³+b³+c³+3.(a+b).(b+c).(c+a)

d) (a+b).(b+c).(c+a)=(a+b+c).(ab+bc+ca)-abc

e) ab.(a+b)-bc.(b+c)+ac.(a-c)=(a+b).(b+c).(a-c)

f) 2bc.(b+2c)+2a.(c-2a)-2ab.(a+2b-7abc)= (b+2c).(c-2a).(a+2b)

Answer 1

a) Biến đổi VT ta có :

(a²-b²)² + (2ab)²

= a⁴ -2a²+b⁴+4a²b²

= a⁴+2a²b² +b⁴

= (a²b²)² = VP (đpcm)

Answer 2

b) Biến đổi vế trái ta có :

(ax+b)² + (a-bx)²+cx²+c²

= a²x²+2axb+b² +a² - 2axb+b²x² +c²x²+ c²

= (a²+b²+c²) + x²(a²+b²+c²)

= (a²+b²+c²) (x²+1) = VP (đpcm)

Answer 3

a) ta có : VT = \(\left(a^2-b^2\right)^2+\left(2ab\right)^2\) = \(a^4-2\left(ab\right)^2+b^4+4\left(ab\right)^2\)

= \(\left(a^2+b^2\right)^2\) = VP (đpcm)

b) ta có : VT = \(\left(ax+b\right)^2+\left(a-bx\right)^2+c^2x^2+c^2\)

= \(\left(ax\right)^2+2axb+b^2+a^2-2abx+\left(bx\right)^2+c^2x^2+c^2\)

= \(\left(ax\right)^2+b^2+a^2+\left(bx\right)^2+\left(cx\right)^2+c^2\)

= \(\left(ax\right)^2+\left(bx\right)^2+\left(cx\right)^2+a^2+b^2+c^2\)

= \(x^2\left(a^2+b^2+c^2\right)+1\left(a^2+b^2+c^2\right)\)

= \(\left(a^2+b^2+c^2\right)\left(x^2+1\right)\) = VP (đpcm)