Cho đa thức \(H\left( x \right) = {x^4} - 3{x^3} - x + 1\). Tìm đa thức P(x) và Q(x) sao cho
a)\(H\left( x \right) + P\left( x \right) = {x^5} - 2{x^2} + 2\)
b)\(H\left( x \right) - Q\left( x \right) = - 2{x^3}\)
a)\(P\left( x \right) = \left( {{x^5} - 2{x^2} + 2} \right) - H\left( x \right)\)
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b)\(Q\left( x \right) = H\left( x \right) - \left( { - 2{x^3}} \right)\)
a)
\(\begin{array}{l}H\left( x \right) + P\left( x \right) = {x^5} - 2{x^2} + 2\\ \Rightarrow P\left( x \right) = \left( {{x^5} - 2{x^2} + 2} \right) - H\left( x \right)\\ \Rightarrow P\left( x \right) = {x^5} - 2{x^2} + 2 - \left( {{x^4} - 3{x^3} - x + 1} \right)\\ \Rightarrow P\left( x \right) = {x^5} - 2{x^2} + 2 - {x^4} + 3{x^3} + x - 1\\ \Rightarrow P\left( x \right) = {x^5} - {x^4} + 3{x^3} + x + 1\end{array}\)
b)
\(\begin{array}{l}H\left( x \right) - Q\left( x \right) = - 2{x^3}\\ \Rightarrow Q\left( x \right) = H\left( x \right) - \left( { - 2{x^3}} \right)\\ \Rightarrow Q\left( x \right) = \left( {{x^4} - 3{x^3} - x + 1} \right) + 2{x^3}\\ \Rightarrow Q\left( x \right) = {x^4} - {x^3} - x + 1\end{array}\)