Question

Answer 1

a: \(P\left(x\right)=-7x^3+5x^4+x^2+5+3x^3-4x^4+4x^3-4\)

\(=\left(5x^4-4x^4\right)+\left(-7x^3+3x^3+4x^3\right)+x^2+1=x^4+x^2+1\)

b: \(P\left(\frac12\right)=\left(\frac12\right)^4+\left(\frac12\right)^2+1=\frac{1}{16}+\frac14+1=\frac{1}{16}+\frac{4}{16}+\frac{16}{16}=\frac{21}{16}\)

\(P\left(-1\right)=\left(-1\right)^4+\left(-1\right)^2+1=1+1+1=3\)

c: \(x^4\ge0\forall x;x^2\ge0\forall x\)

Do đó: \(x^4+x^2\ge0\forall x\)

=>\(x^4+x^2+1\ge1>0\forall x\)

=>P(x) không có nghiệm