Bài 1
e) Ta có
\begin{align*}
E &= (x-y-a+2)(x+y+a+2)+(a+y)^2 - (x-2)^2\\
&= [x+2 -(y+a)][(x+2) + (y+a)] +(a+y)^2 - (x-2)^2\\
&= (x+2)^2 - (y+a)^2 +(a+y)^2 - (x-2)^2\\
&= 0
\end{align*}
f) Ta có
\begin{align*}
F&= (x-1)^2 (x+1)^2 (x^2+1)^2 (x^4+1)^2 -(x^8+1)^2\\
&= [(x-1)(x+1)(x^2+1)(x^4+1)]^2 - (x^8+1)^2\\
&= [(x^2-1)(x^2+1)(x^4+1)]^2 - (x^8+1)^2\\
&= [(x^4-1)(x^4+1)]^2 - (x^8+1)^2\\
&=(x^8-1)^2 - (x^8-1)^2 = 0
\end{align*}
Bài 2
c) Ta có
\begin{align*}
C&= x^3 + 12x^2 + 48 x + 63\\
&= x^3 + 3.x^2.4 + 3.x.16 + 64 - 1\\
&= (x^3 + 3x^2.4 + 3.x.4^2 + 4^3) -1\\
&= (x+4)^3-1
\end{align*}
Thay $x = 996$ vào biểu thức ta có
$$C = 1000^3-1 = (10^3)^3-1 = 10^9-1$$
d) Ta có
\begin{align*}
D &= 1999.1995 -1998.1996\\
&= (1998+1)(1996-1) - 1998.1996\\
&= 1998.1996 -1998 + 1996-1 - 1998.1996\\
&= -2-1 = -3
\end{align*}
e) Ta có
\begin{align*}
E &= 2015^2 - 2011.2019\\
&= 2015^2 - (2015-4)(2015+4)\\
&= 2015^2 - (2015^2 - 4^2)\\
&= 2015^2 - 2015^2 + 16\\
&= 16
\end{align*}
