PTĐTTNT theo dạng thêm bớt khi số mũ chia 3 dư 1 chia 3 dư 2 1,$x^{11}$ +$x^{4}$ +1 2,$x^{11}$+ $x^{7}$ +1

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1. `1) x^11+x^4+1`

   `=x^11+ x^10 + x^9 – x^10 – x^9 – x^8 +x^8 + x^7 + x^6 – x^7 – x^6 – x^5  + x^5 + x^4 + x^3 – x^3 + 1.`

   `=(x^11+ x^10 + x^9) – (x^10 + x^9 + x^8) +(x^8 + x^7 + x^6) – (x^7 + x^6 + x^5)  + (x^5 + x^4 + x^3) – (x^3 – 1).`

   `= x^9(x^2+x+1)  -x^8(x^2+x+1)  +x^6(x^2+x+1) -x^5(x^2+x+1) + x^3(x^2+x+1) – (x-1)(x^2+x+1).`

   `=(x^2+x+1)(x^9-x^8+x^6-x^5+x^3 – x +1 ).`

   `2) x^11 + x^7 + 1`

   `= x^11 + x^10 + x^9 – x^10 – x^9 – x^8 + x^8 +  x^7 + x^6 – x^6  – x^5 – x^4 + x^5 + x^4 + x^3 – x^3 + 1`

   `= (x^11 + x^10 + x^9) – ( x^10 + x^9 + x^8) + (x^8 +  x^7 + x^6) – ( x^6 + x^5+ x^4) +( x^5 + x^4 + x^3) – (x^3 – 1)`

   `= x^9(x^2+x+1) -x^8(x^2+x+1) + x^6(x^2+x+1) – x^4(x^2+x+1) + x^3(x^2+x+1) – (x-1)(x^2+x+1).`

   `= (x^2+x+1)(x^9-x^8+x^6-x^4+x^3-x+1).`

    

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2. $x^{11} + x^{4} + 1$

   $= (x^{11} + x^{10} + x^{9}) – (x^{10} + x^{9} + x^{8}) + (x^{8} + x^{7} + x^{6}) – (x^{7} + x^{6} + x^{5}) + (x^{5} + x^{4} + x^{3}) – (x^{3} + x^{2} + x) + (x^{2} + x + 1)$

   $= x^{9}(x^{2} + x + 1) – x^{8}(x^{2} + x + 1) + x^{6}(x^{2} + x + 1) – x^{5}(x^{2} + x + 1) + x^{3}(x^{2} + x + 1) – x^{2}(x^{2} + x + 1) + (x^{2} + x + 1)$

   $= (x^{2} + x + 1)(x^{9} – x^{8} + x^{6} – x^{5} + x^{3} – x^{2} + 1)$

   $x^{11} + x^{7} + 1$

   $= (x^{11} + x^{10} + x^{9}) – (x^{10} + x^{9} + x^{8}) + (x^{8} + x^{7} + x^{6}) – (x^{6} + x^{5} + x^{4}) + (x^{5} + x^{4} + x^{3}) – (x^{4} + x^{3} + x^{2}) + (x^{2} + x + 1)$

   $= x^{9}(x^{2} + x + 1) – x^{8}(x^{2} + x + 1) + x^{6}(x^{2} + x + 1) – x^{4}(x^{2} + x + 1) + x^{3}(x^{2} + x + 1) – x^{2}(x^{2} + x + 1) + (x^{2} + x + 1)$

   $= (x^{2} + x + 1)(x^{9} – x^{8} + x^{6} – x^{4} + x^{3} – x^{2} + 1)$

    

    

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