قم بتحليل التعابير الجبرية التالية إلى عواملها الأولية باستخدام التحليل بالتجميع
F
\( F = 10(2x - 1)^2 + 5(2x - 1) \)
\[= 10(4x^2 - 4x + 1) + 5(2x - 1)\]
\[= 40x^2 - 40x + 10 + 10x - 5\]
\[= 40x^2 - 30x + 5\]
G
\( G = (x + \sqrt{2})(2x - 5) - (x + \sqrt{2})(3x - 1) \)
\[= (x + \sqrt{2})[(2x - 5) - (3x - 1)]\]
\[= (x + \sqrt{2})(-x - 4)\]
\[= -x(x + \sqrt{2}) - 4(x + \sqrt{2})\]
\[= -x^2 - x\sqrt{2} - 4x - 4\sqrt{2}\]
H
\( H = 2(x + 1)(2x - \sqrt{3}) - 3(x + 1)(x - 2\sqrt{3}) \)
\[= (x + 1)[2(2x - \sqrt{3}) - 3(x - 2\sqrt{3})]\]
\[= (x + 1)[4x - 2\sqrt{3} - 3x + 6\sqrt{3}]\]
\[= (x + 1)(x + 4\sqrt{3})\]
I
\( I = (x + 1)(2x - 3) - x(2x + 2) \)
\[= (x + 1)(2x - 3) - x(2x + 2)\]
\[= (x + 1)(2x - 3) - 2x(x + 1)\]
\[= (x + 1)[(2x - 3) - 2x]\]
\[= (x + 1)(-3)\]
\[= -3(x + 1)\]
J
\( J = (3x - 6)(x - 3) + (x + 1)(4x - 8) \)
\[= 3(x - 2)(x - 3) + 4(x + 1)(x - 2)\]
\[= (x - 2)[3(x - 3) + 4(x + 1)]\]
\[= (x - 2)[3x - 9 + 4x + 4]\]
\[= (x - 2)(7x - 5)\]
K
\( K = a(x + \sqrt{3}) - x - \sqrt{3} \)
\[= a(x + \sqrt{3}) - 1(x + \sqrt{3})\]
\[= (a - 1)(x + \sqrt{3})\]
L
\( L = x(x - \sqrt{3}) + (\sqrt{3}x - 3) \)
\[= x(x - \sqrt{3}) + \sqrt{3}x - 3\]
\[= x(x - \sqrt{3}) + \sqrt{3}(x - \sqrt{3})\]
\[= (x - \sqrt{3})(x + \sqrt{3})\]
M
\( M = x(a - \sqrt{5}) + (\sqrt{2}a - \sqrt{10}) \)
\[= x(a - \sqrt{5}) + \sqrt{2}(a - \sqrt{5})\]
\[= (a - \sqrt{5})(x + \sqrt{2})\]
N
\( N = \sqrt{3}(x - \sqrt{3}) - \sqrt{6} + x\sqrt{2} \)
\[= \sqrt{3}(x - \sqrt{3}) + x\sqrt{2} - \sqrt{6}\]
\[= x\sqrt{3} - 3 + x\sqrt{2} - \sqrt{6}\]
\[= x(\sqrt{3} + \sqrt{2}) - (3 + \sqrt{6})\]
\[= x(\sqrt{3} + \sqrt{2}) - \sqrt{3}(\sqrt{3} + \sqrt{2})\]
\[= (\sqrt{3} + \sqrt{2})(x - \sqrt{3})\]