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Question Number 206037 by Davidtim last updated on 05/Apr/24
is it a polynomial?  x^3 −2x^2 +(√x^2 )+10
$${is}\:{it}\:{a}\:{polynomial}? \\ $$$${x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +\sqrt{{x}^{\mathrm{2}} }+\mathrm{10} \\ $$
Answered by Frix last updated on 05/Apr/24
f(x)=x^3 −2x^2 +(√x^2 )+10  x∈R ⇒ (√x^2 )=∣x∣  ⇒  f(x)= { ((x^3 −2x^2 −x+10; x<0)),((x^3 −2x^2 +x+10; x≥0)) :}  This is not a polynomial.
$${f}\left({x}\right)={x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +\sqrt{{x}^{\mathrm{2}} }+\mathrm{10} \\ $$$${x}\in\mathbb{R}\:\Rightarrow\:\sqrt{{x}^{\mathrm{2}} }=\mid{x}\mid \\ $$$$\Rightarrow \\ $$$${f}\left({x}\right)=\begin{cases}{{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} −{x}+\mathrm{10};\:{x}<\mathrm{0}}\\{{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +{x}+\mathrm{10};\:{x}\geqslant\mathrm{0}}\end{cases} \\ $$$$\mathrm{This}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{polynomial}. \\ $$
Answered by BaliramKumar last updated on 05/Apr/24
yes
$$\mathrm{yes} \\ $$

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