Question

Answer 1

a) \(P\left(x\right)=3x^2-4x^4-5x+9+6x^4+2x^3-5\)

\(P\left(x\right)=\left(-4x^4+6x^4\right)+2x^3+3x^2-5x+\left(9-5\right)\\ P\left(x\right)=2x^4+2x^3+3x^2-5x+4\)

 

\(Q\left(x\right)=-3x^3+x-x^4-3+x^3+4x-2x^2\\ Q\left(x\right)=-x^4+\left(-3x^3+x^3\right)-2x^2+\left(x+4x\right)-3\\ Q\left(x\right)=-x^4-2x^3-2x^2+5x-3\)

b) \(M\left(x\right)=P\left(x\right)+Q\left(x\right)\)

\(M\left(x\right)=\left(2x^4-x^4\right)+\left(2x^3-2x^3\right)+\left(3x^2-2x^2\right)+\left(-5x+5x\right)+\left(4-3\right)\\ M\left(x\right)=x^4+x^2+1\)

c) Ta có \(x^4\ge0;x^2\ge0\forall x\Rightarrow x^4+x^2+1>0\Rightarrow M\left(x\right)>0,\forall x\)