Question

Tìm các đa thức P(x) và Q(x), biết rằng:

P(x) + Q(x) = x^2+ 1

P(x) –Q(x) = 2x

Answer 1

P(x) + Q(x) = x^2 + 1 (1) 

P(x) - Q(x) = 2x (2) 

Lấy (1) + (2) suy ra : \(2P\left(x\right)=x^2+2x+1=\left(x+1\right)^2\Rightarrow P\left(x\right)=\dfrac{\left(x+1\right)^2}{2}\)

\(Q\left(x\right)=P\left(x\right)-2x=\dfrac{\left(x+1\right)^2-4x}{2}=\dfrac{\left(x-1\right)^2}{2}\)

Answer 2

\(P\left(x\right)+Q\left(x\right)+P\left(x\right)-Q\left(x\right)=x^2+1+2x\\ \Rightarrow2P\left(x\right)=x^2+2x+1\\ \Rightarrow P\left(x\right)=\dfrac{x^2+2x+1}{2}\)

\(P\left(x\right)-Q\left(x\right)=2x\\ \Leftrightarrow\dfrac{x^2+2x+1}{2}-Q\left(x\right)=2x\\ \Leftrightarrow Q\left(x\right)=\dfrac{x^2+2x+1}{2}-2x\\ \Leftrightarrow Q\left(x\right)=\dfrac{x^2+2x+1-4x}{2}\\ \Leftrightarrow Q\left(x\right)=\dfrac{x^2-2x+1}{2}\)