Step 1

Given:

\(\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{3}}}}{{{2}}}}−{x}^{{2}}+{2}\)

Step 2

Convert element to fracnion: \(\displaystyle{x}^{{2}}={\frac{{{x}^{{{2}}}{2}}}{{{2}}}},{2}={\frac{{{2}\times{2}}}{{{2}}}}\)

\(\displaystyle={\frac{{{x}^{{3}}}}{{{2}}}}-{\frac{{{x}^{{2}}\times{2}}}{{{2}}}}+{\frac{{{2}\times{2}}}{{{2}}}}\)

Since the denominators are equal, combine the fractions: \(\displaystyle{\frac{{{a}}}{{{c}}}}\pm{\frac{{{b}}}{{{c}}}}={\frac{{{a}\pm{b}}}{{{c}}}}\)

\(\displaystyle={\frac{{{x}^{{3}}-{x}^{{2}}\times{2}+{2}\times{2}}}{{{2}}}}\)

Multiply the numbers: \(\displaystyle{2}\times{2}={4}\)

\(\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{3}}-{2}{x}^{{2}}+{4}}}{{{2}}}}\)

Hence, it is a polynomial with its degree 3,

Step 3

Now, convert 3 degree polynomial into standard form which is

\(\displaystyle{a}{x}^{{{3}}}+{b}{x}^{{{3}}}+{c}{x}+{d}\)

Standard form \(\displaystyle\rightarrow{g{{\left({x}\right)}}}={\frac{{{x}^{{3}}}}{{{2}}}}-{2}{x}^{{{2}}}+{0}{x}+{2}\)

Leading term \(\displaystyle\rightarrow{\frac{{{1}}}{{{2}}}}\)

Constant term \(\displaystyle\rightarrow{2}\)

Given:

\(\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{3}}}}{{{2}}}}−{x}^{{2}}+{2}\)

Step 2

Convert element to fracnion: \(\displaystyle{x}^{{2}}={\frac{{{x}^{{{2}}}{2}}}{{{2}}}},{2}={\frac{{{2}\times{2}}}{{{2}}}}\)

\(\displaystyle={\frac{{{x}^{{3}}}}{{{2}}}}-{\frac{{{x}^{{2}}\times{2}}}{{{2}}}}+{\frac{{{2}\times{2}}}{{{2}}}}\)

Since the denominators are equal, combine the fractions: \(\displaystyle{\frac{{{a}}}{{{c}}}}\pm{\frac{{{b}}}{{{c}}}}={\frac{{{a}\pm{b}}}{{{c}}}}\)

\(\displaystyle={\frac{{{x}^{{3}}-{x}^{{2}}\times{2}+{2}\times{2}}}{{{2}}}}\)

Multiply the numbers: \(\displaystyle{2}\times{2}={4}\)

\(\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{3}}-{2}{x}^{{2}}+{4}}}{{{2}}}}\)

Hence, it is a polynomial with its degree 3,

Step 3

Now, convert 3 degree polynomial into standard form which is

\(\displaystyle{a}{x}^{{{3}}}+{b}{x}^{{{3}}}+{c}{x}+{d}\)

Standard form \(\displaystyle\rightarrow{g{{\left({x}\right)}}}={\frac{{{x}^{{3}}}}{{{2}}}}-{2}{x}^{{{2}}}+{0}{x}+{2}\)

Leading term \(\displaystyle\rightarrow{\frac{{{1}}}{{{2}}}}\)

Constant term \(\displaystyle\rightarrow{2}\)