Toán Lớp 8: chuyển thành nhân tử
e, x^(10)+x^5+1
f, x^8+x+1
g, x^8+x^4+1
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Toán Lớp 8: chuyển thành nhân tử
e, x^(10)+x^5+1
f, x^8+x+1
g, x^8+x^4+1
Comments ( 1 )
e,\\
{x^{10}} + {x^5} + 1 = \left( {{x^2} + x + 1} \right).\left( {{x^8} – {x^7} + {x^5} – {x^4} + {x^3} – x + 1} \right)\\
f,\\
{x^8} + x + 1 = \left( {{x^2} + x + 1} \right).\left( {{x^6} – {x^5} + {x^3} – {x^2} + 1} \right)\\
g,\\
{x^8} + {x^4} + 1 = \left( {{x^2} + x + 1} \right).\left( {{x^6} – {x^5} + {x^3} – x + 1} \right)
\end{array}\)
e,\\
{x^{10}} + {x^5} + 1\\
= \left( {{x^{10}} – x} \right) + \left( {{x^5} – {x^2}} \right) + \left( {{x^2} + x + 1} \right)\\
= x.\left( {{x^9} – 1} \right) + {x^2}.\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= x.\left[ {{{\left( {{x^3}} \right)}^3} – {1^3}} \right] + {x^2}.\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= x.\left( {{x^3} – 1} \right).\left[ {{{\left( {{x^3}} \right)}^2} + {x^3}.1 + {1^2}} \right] + {x^2}.\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= x\left( {{x^3} – 1} \right).\left( {{x^6} + {x^3} + 1} \right) + {x^2}.\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^3} – 1} \right).\left( {{x^7} + {x^4} + x} \right) + {x^2}.\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^3} – 1} \right).\left( {{x^7} + {x^4} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {x – 1} \right).\left( {{x^2} + x + 1} \right).\left( {{x^7} + {x^4} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^2} + x + 1} \right).\left[ {\left( {x – 1} \right).\left( {{x^7} + {x^4} + {x^2} + x} \right) + 1} \right]\\
= \left( {{x^2} + x + 1} \right).\left( {{x^8} + {x^5} + {x^3} + {x^2} – {x^7} – {x^4} – {x^2} – x + 1} \right)\\
= \left( {{x^2} + x + 1} \right).\left( {{x^8} – {x^7} + {x^5} – {x^4} + {x^3} – x + 1} \right)\\
f,\\
{x^8} + x + 1\\
= \left( {{x^8} – {x^2}} \right) + {x^2} + x + 1\\
= {x^2}.\left( {{x^6} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= {x^2}.\left[ {{{\left( {{x^3}} \right)}^2} – 1} \right] + \left( {{x^2} + x + 1} \right)\\
= {x^2}.\left( {{x^3} – 1} \right).\left( {{x^3} + 1} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^3} – 1} \right).\left( {{x^5} + {x^2}} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {x – 1} \right).\left( {{x^2} + x + 1} \right).\left( {{x^5} + {x^2}} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^2} + x + 1} \right).\left[ {\left( {x – 1} \right).\left( {{x^5} + {x^2}} \right) + 1} \right]\\
= \left( {{x^2} + x + 1} \right).\left( {{x^6} + {x^3} – {x^5} – {x^2} + 1} \right)\\
= \left( {{x^2} + x + 1} \right).\left( {{x^6} – {x^5} + {x^3} – {x^2} + 1} \right)\\
g,\\
{x^8} + {x^4} + 1\\
= \left( {{x^8} – {x^2}} \right) + \left( {{x^4} – x} \right) + \left( {{x^2} + x + 1} \right)\\
= {x^2}.\left( {{x^6} – 1} \right) + x\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= {x^2}.\left[ {{{\left( {{x^3}} \right)}^2} – {1^2}} \right] + x.\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= {x^2}.\left( {{x^3} – 1} \right)\left( {{x^3} + 1} \right) + x.\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^3} – 1} \right).\left( {{x^5} + {x^2}} \right) + x.\left( {{x^3} – 1} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^3} – 1} \right).\left( {{x^5} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {x – 1} \right).\left( {{x^2} + x + 1} \right)\left( {{x^5} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^2} + x + 1} \right).\left[ {\left( {x – 1} \right)\left( {{x^5} + {x^2} + x} \right) + 1} \right]\\
= \left( {{x^2} + x + 1} \right).\left( {{x^6} + {x^3} + {x^2} – {x^5} – {x^2} – x + 1} \right)\\
= \left( {{x^2} + x + 1} \right).\left( {{x^6} – {x^5} + {x^3} – x + 1} \right)
\end{array}\)