Question

Cho đa thức \(f\left(x\right)=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0\)

Biết rằng: \(f\left(1\right)=f\left(-1\right);f\left(2\right)=f\left(-2\right)\)

Chứng minh: \(f\left(x\right)=f\left(-x\right)\forall x\)

Answer 1

Lời giải:

\(f(1)=f(-1)\)

\(\Leftrightarrow a_4+a_3+a_2+a_1+a_0=a_4-a_3+a_2-a_1+a_0\)

\(\Leftrightarrow 2(a_3+a_1)=0\Leftrightarrow a_3+a_1=0(1)\)

\(f(2)=f(-2)\)

\(\Leftrightarrow 16a_4+8a_3+4a_2+2a_1+a_0=16a_4-8a_3+4a_2-2a_1+a_0\)

\(\Leftrightarrow 16a_3+4a_1=0\Leftrightarrow 4a_3+a_1=0(2)\)

Từ \((1);(2)\Rightarrow a_3=a_1=0\)

Do đó:
\(f(x)=a_4x^4+a_2x^2+a_0\)

\(\Rightarrow f(-x)=a_4(-x)^4+a_2(-x)^2+a_0=a_4x^4+a_2x^2+a_0\)

Vậy $f(x)=f(-x)$.