Cho hai đa thức:

\(M\left( x \right) = 2 - 5{x^2} + 3{x^4} - 4{x^2} + 3x + {x^4} - 4{x^6} - 7x\)

\(N\left( x \right) = {\rm{ \;}} - 1 + 5{x^6} - 6{x^2} - 5 - 9{x^6} + 4{x^4} - 3{x^2}\)

a) Thu gọn và sắp xếp các đa thức theo lũy thừa giảm dần của biến.

b) Tìm đa thức \(H\left( x \right)\) và \(G\left( x \right)\) biết \(H\left( x \right) = M\left( x \right) + N\left( x \right)\) và \(G\left( x \right) = M\left( x \right) - N\left( x \right)\).

c) Tìm nghiệm của đa thức \(G\left( x \right)\).

 

\(M\left( x \right) = 2 - 5{x^2} + 3{x^4} - 4{x^2} + 3x + {x^4} - 4{x^6} - 7x\)

\(N\left( x \right) = {\rm{ \;}} - 1 + 5{x^6} - 6{x^2} - 5 - 9{x^6} + 4{x^4} - 3{x^2}\)

a) Ta có:

\(\begin{array}{*{20}{l}}{M\left( x \right) = 2 - 5{x^2} + 3{x^4} - 4{x^2} + 3x + {x^4} - 4{x^6} - 7x}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = {\rm{ \;}} - 4{x^6} + \left( {3{x^4} + {x^4}} \right) + \left( { - 5{x^2} - 4{x^2}} \right) + \left( {3x - 7x} \right) + 2}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = {\rm{ \;}} - 4{x^6} + 4{x^4} - 9{x^2} - 4x + 2}\end{array}\)

\(\begin{array}{*{20}{l}}{N\left( x \right) = {\rm{ \;}} - 1 + 5{x^6} - 6{x^2} - 5 - 9{x^6} + 4{x^4} - 3{x^2}}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = \left( {5{x^6} - 9{x^6}} \right) + 4{x^4} + \left( { - 6{x^2} - 3{x^2}} \right) + \left( { - 1 - 5} \right)}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = {\rm{ \;}} - 4{x^6} + 4{x^4} - 9{x^2} - 6}\end{array}\)

b) Ta có:

\(\begin{array}{*{20}{l}}{H\left( x \right) = M\left( x \right) + N\left( x \right)}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = ( - 4{x^6} + 4{x^4} - 9{x^2} - 4x + 2) + ( - 4{x^6} + 4{x^4} - 9{x^2} - 6)}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = \left( { - 4{x^6} - 4{x^6}} \right) + \left( {4{x^4} + 4{x^4}} \right) + \left( { - 9{x^2} - 9{x^2}} \right) - 4x + \left( {2 - 6} \right)}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = {\rm{ \;}} - 8{x^6} + 8{x^4} - 18{x^2} - 4x - 4}\end{array}\)

\(\begin{array}{*{20}{l}}{G\left( x \right) = M\left( x \right) - N\left( x \right)}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = ( - 4{x^6} + 4{x^4} - 9{x^2} - 4x + 2) - \left( { - 4{x^6} + 4{x^4} - 9{x^2} - 6} \right)}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = {\rm{ \;}} - 4{x^6} + 4{x^4} - 9{x^2} - 4x + 2 + 4{x^6} - 4{x^4} + 9{x^2} + 6}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = \left( { - 4{x^6} + 4{x^6}} \right) + \left( {4{x^4} - 4{x^4}} \right) + \left( { - 9{x^2} + 9{x^2}} \right) - 4x + \left( {2 + 6} \right)}\\{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}  = {\rm{ \;}} - 4x + 8}\end{array}\)

c) \(G\left( x \right) = 0 \Rightarrow {\rm{ \;}} - 4x + 8 = 0 \Rightarrow {\rm{ \;}} - 4x = {\rm{ \;}} - 8 \Rightarrow x = 2\).