Rút gọn các biểu thực sau:
$$

$a) {(a + b)}^2 – {(a – b)}^2$

$$ b) {(a+b)}^3-{(a-b)}^3-2b^3$$ $$

$$ c) {(x+y+z)}^2-2(x+y+z\left)\right(x+y)+{(x+y)}^2$$

Giải:

$a) {(a + b)}^2 – {(a – b)}^2$

$= (a^2 + 2ab + b^2) – (a^2 – 2ab + b^2)$

$= a^2 + 2ab + b^2 – a^2 + 2ab - b^2 = 4ab$

Hoặc :

${(a + b)}^2 – {(a – b)}^2 = \left[\right(a + b) + (a – b\left)\right]\left[\right(a + b) – (a – b\left)\right]$

 $= (a + b + a – b)(a + b – a + b)$

$= 2a . 2b = 4ab$

$b) {(a+b)}^3-{(a-b)}^3-2b^3$${\mathrm{}}^{}$

$= (a^3 + 3a^{2}b + 3a{b}^2 + b^3) – (a^3 – 3a^{2}b + 3a{b}^2 – b^3) – 2b^3$

$= a^3 + 3a^{2}b + 3a{b}^2 + b^3 – a^3 + 3a^{2}b - 3a{b}^2 + b^3 – 2b^3$

$= 6a^{2}b$

Hoặc :

${(a + b)}^3 – {(a – b)}^3 – 2b^3 = [{(a + b)}^3 – {(a – b)}^3] – 2b^3$

$= \left[\right(a + b) – (a – b\left)\right][{(a + b)}^2 + (a + b\left)\right(a – b) + {(a – b)}^2] – 2b^3$

$= (a + b – a + b)(a^2+ 2ab + b^2 + a^2 – b^2 + a^2 – 2ab + b^2) – 2b^3$

$= 2b . (3a^2 + b^2) – 2b^3= 6a^{2}b + 2b^3 – 2b^3 = 6a^{2}b$

$c) {(x+y+z)}^2-2(x+y+z\left)\right(x+y)+{(x+y)}^2$${\mathrm{}}^{}$

$= x^2+y^2+z^2+ 2xy + 2yz + 2xz – 2(x^2 + xy + yx + y^2 + zx + zy) + x^2 + 2xy + y^2$

$= 2x^2+2y^2 + z^2 + 4xy + 2yz + 2xz – 2x^2 – 4xy – 2y^2 – 2xz – 2yz = z^2$