Bài 1: phân tích đa thức thành nhân tử a) x mũ 4+16 b) x mũ 4.y mũ 4+64 c) x mũ 4. y mũ 4+4 d) 4x mũ 4y mũ 4+1 e) x mũ 4+ 1 f) x mũ 8+x+1 g) x mũ 8+x

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1. Đáp án:

    $a) (x^2+4-\sqrt{8}x)(x^2+4+\sqrt{8}x)\\
   b)
   (x^2y^2+8-4xy)(x^2y^2+8+4xy)\\
   c)
   (x^2y^2+4-2xy)(x^2y^2+4+2xy)\\
   d)
   (2x^2y^2+1-2xy)(2x^2y^2+1+2xy)\\
   e)
   (x^2+1-\sqrt{2}x)(x^2+1+\sqrt{2}x)\\
   f) 
   (x^2+x+1)(x^6-x^5+x^3-x^2+1)\\
   g)
   (x^2+x+1)(x^6-x^4+x^3-x+1)\\
   h)
   (x^4+1-x^2)(x^4+1+x^2)\\
   k)
   (x^2+2y^2-2xy)(x^2+2y^2+2xy)$

   Giải thích các bước giải:

    $a) x^4+16\\
   =x^4+16+8x^2-8x^2\\
   =(x^4+16+8x^2)-8x^2\\
   =\left [(x^2)^2 +2.x^2.4+4^2 \right ]-(\sqrt{8}x)^2\\
   =(x^2+4)^2-(\sqrt{8}x)^2\\
   =(x^2+4-\sqrt{8}x)(x^2+4+\sqrt{8}x)\\
   b)
   x^4y^4+64\\
   =(x^2y^2)^2+16x^2y^2+8^2-16x^2y^2\\
   =(x^2y^2+8)^2-(4xy)^2\\
   =(x^2y^2+8-4xy)(x^2y^2+8+4xy)\\
   c)
   x^4y^4+4\\
   =(x^2y^2)^2+4x^2y^2+2^2-4x^2y^2\\
   =(x^2y^2+4)^2-(2xy)^2\\
   =(x^2y^2+4-2xy)(x^2y^2+4+2xy)\\
   d)
   4x^4y^4+1\\
   =(2x^2y^2)^2+2.2x^2y^2.1+1-4x^2y^2\\
   =(2x^2y^2+1)^2-(2xy)^2\\
   =(2x^2y^2+1-2xy)(2x^2y^2+1+2xy)\\
   e)
   x^4+1\\
   =(x^2)^2+2x^2+1-2x^2\\
   =(x^2+1)^1-(\sqrt{2}x)^2\\
   =(x^2+1-\sqrt{2}x)(x^2+1+\sqrt{2}x)\\
   f) 
   x^8+x+1\\
   =x^8+x^7-x^7+x^6-x^6+x^5-x^5+x^4-x^4+x^3-x^3+x^2-x^2+x+1\\
   =(x^8+x^7+x^6)-(x^7+x^6+x^5)+(x^5+x^4+x^3)-(x^4+x^3+x^2)+(x^2+x+1)\\
   =x^6(x^2+x+1)-x^5(x^2+x+1)+x^3(x^2+x+1)-x^2(x^2+x+1)+(x^2+x+1)\\
   =(x^2+x+1)(x^6-x^5+x^3-x^2+1)\\
   g)
   x^8+x^7+1\\
   =x^8+x^7+x^6-x^6+x^5-x^5+x^4-x^4+x^3-x^3+x^2-x^2+x-x+1\\
   =(x^8+x^7+x^6)-(x^6+x^5+x^4)+(x^5+x^4+x^3)-(x^3+x^2+x)+(x^2+x+1)\\
   =x^6(x^2+x+1)-x^4(x^2+x+1)+x^3(x^2+x+1)-x(x^2+x+1)+(x^2+x+1)\\
   =(x^2+x+1)(x^6-x^4+x^3-x+1)\\
   h)
   x^8+x^4+1\\
   =x^8+2x^4+1-x^4\\
   =(x^4+1)^2-(x^2)^2\\
   =(x^4+1-x^2)(x^4+1+x^2)\\
   k)
   x^4+4y^4\\
   =(x^2)^2+2.x^2.2y^2+(2y^2)^2-4x^2y^2\\
   =(x^2+2y^2)^2-(2xy)^2\\
   =(x^2+2y^2-2xy)(x^2+2y^2+2xy)$

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