Calculate the left side and the right side. \[2\times(12+3)\quad\quad 2\times 12 + 2\times 3\] \[3\times(0.8+1)\quad\quad 3\times 0.8 + 3\times 1\] \[(-2)\times (3-5)\quad\quad (-2)\times 3 + (-2)\times (-5)\] What do you notice?
Show that for any number \(a,b,c,d\), we have \((a+b)(c+d) = ac + ad + bc + bd\).
Expand \((x_1+\dots + x_n)^2\) where \(x_1,\dots,x_n\) are real numbers.
Prove the Cauchy-Schwarz inequality \[(a_1b_1+\dots+a_nb_n)^2\leq (a_1^2+\dots+a_n^2)(b_1^2+\dots+b_n^2)\] where \(a_1,\dots,a_n,b_1,\dots,b_n\) are real numbers. If you already know a proof (or more!), find a new one.