Phân tích đa thức thành nhân tử :
a) x^4 + 5x^3 + 10x – 4
b) x^4 +y^4 +(x+y) ^4
c) (x-y)^3 + (y -z) ^3 + (z – x) ^3
d) x^4+ 7x^2 – 6x + 1
Phân tích đa thức thành nhân tử : a) x^4 + 5x^3 + 10x – 4 b) x^4 +y^4 +(x+y) ^4 c) (x-y)^3 + (y -z) ^3 + (z – x) ^3 d) x^4+ 7x^2 – 6x + 1
By Sarah
\[\begin{array}{l}
c)\,\,{\left( {x – y} \right)^3} + {\left( {y – z} \right)^3} + {\left( {z – x} \right)^3}\\
= \left( {x – y + y – z} \right)\left[ {{{\left( {x – y} \right)}^2} – \left( {x – y} \right)\left( {y – z} \right) + {{\left( {y – z} \right)}^2}} \right] – {\left( {x – z} \right)^3}\\
= \left( {x – z} \right)\left[ {{{\left( {x – y} \right)}^2} – \left( {x – y} \right)\left( {y – z} \right) + {{\left( {y – z} \right)}^2} – {{\left( {x – z} \right)}^2}} \right]\\
= \left( {x – z} \right)\left\{ {\left[ {{{\left( {x – y} \right)}^2} – {{\left( {x – z} \right)}^2}} \right] – \left[ {\left( {x – y} \right)\left( {y – z} \right) + {{\left( {y – z} \right)}^2}} \right]} \right\}\\
= \left( {x – z} \right)\left[ {\left( {x – y – x + z} \right)\left( {x – y + x – z} \right) – \left( {y – z} \right)\left( {x – y + y – z} \right)} \right]\\
= \left( {x – z} \right)\left[ {\left( {z – y} \right)\left( {2x + y – z} \right) + \left( {z – y} \right)\left( {x – z} \right)} \right]\\
= \left( {x – z} \right)\left( {z – y} \right)\left( {2x + y – z + x – z} \right)\\
= \left( {x – z} \right)\left( {z – y} \right)\left( {3x + y – 2z} \right).
\end{array}\]
Câu d em đã chép đúng đề bài chưa e?
a) \(x^4+5x^3+10x-4\)
\(=(x^4-4)+(5x^3+10x)\)
\(=(x^2-2)(x^2+2)+5x(x^2+2)\).
b) \(x^4+y^4+[(x+y)^2]^2\)
\(=x^4+y^4+(x^2+y^2+2xy)^2\)
\(=x^4+y^4+x^4+y^4+4x^2y^2+2x^2y^2+4x^3y+4xy^3\)
\(=2x^4+2y^4+6x^2y^2+4x^3y+4xy^3\)
\(=2(x^4+y^4+2x^2y^2)+2x^2y^2+4xy(x^2+y^2)\)
\(=2(x^2+y^2)^2+2x^2y^2+4xy(x^2+y^2)\)
\(=2(x^2+y^2+xy)\)