Question

Chứng minh rằng:

a) a³+b³= (a+b) [(a-b)²+ab]

b) (a²+b²) (c²+d²)= (ac +bd)²+(ad-bc)²

Answer 1

Lời giải :

a) \(VP=\left(a+b\right)\left[\left(a-b\right)^2+ab\right]\)

\(=\left(a+b\right)\left(a^2-2ab+b^2+ab\right)\)

\(=\left(a+b\right)\left(a^2-ab+b^2\right)\)

\(=a^3+b^3=VT\)( đpcm )

b) \(VT=\left(a^2+b^2\right)\left(c^2+d^2\right)\)

\(=a^2c^2+a^2d^2+b^2c^2+b^2d^2\)

\(=a^2c^2+2abcd+b^2d^2+a^2d^2-2abcd+b^2c^2\)

\(=\left(ac+bd\right)^2+\left(ad-bc\right)^2=VP\)( đpcm )

Answer 2

a)CM \(a^3+b^3=\left(a+b\right)\left[\left(a-b\right)^2+ab\right]\)

VT = \(a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)

VP = \(\left(a+b\right)\left[\left(a-b\right)^2+ab\right]=\left(a+b\right)\left(a^2-2ab+b^2+ab\right)=\left(a+b\right)\left(a^2-ab+b^2\right)\)

Ta thấy VP = VT

=> \(a^3+b^3=\left(a+b\right)\left[\left(a-b\right)^2+ab\right]\)

b) CM \(\left(a^2+b^2\right)\left(c^2+d^2\right)=\left(ac+bd\right)^2+\left(ad-bc\right)^2\)

VT = \(\left(a^2+b^2\right)\left(c^2+d^2\right)=a^2c^2+a^2d^2+b^2c^2+b^2d^2\)

VP = \(\left(ac+bd\right)^2+\left(ad-bc\right)^2=ac^2+2acbd+bd^2+ad^2-2abcd+bc^2=ac^2+ad^2+bd^2+bc^2\)Ta thấy VP = VT

=> \(\left(a^2+b^2\right)\left(c^2+d^2\right)=\left(ac+bd\right)^2+\left(ad-bc\right)^2\)