Question

Determine whether g(x) = x^{3}/2 − x^{2} + 2 is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form.

Polynomial arithmetic
Determine whether $$\displaystyle{g{{\left({x}\right)}}}=\frac{{x}^{{{3}}}}{{2}}−{x}^{{{2}}}+{2}$$ is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Identify the leading term and the constant term.

2021-08-04
Step 1
Given:
$$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{{3}}}}}{{{2}}}}-{x}^{{{2}}}+{2}$$
Step 2
$$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{{3}}}}}{{{2}}}}-{x}^{{{2}}}+{2}$$
Convert element to fraction: $$\displaystyle{x}^{{{2}}}={\frac{{{x}^{{{2}}}{2}}}{{{2}}}},{2}={\frac{{{2}\cdot{2}}}{{{2}}}}$$
$$\displaystyle={\frac{{{x}^{{{3}}}}}{{{2}}}}-{\frac{{{x}^{{{2}}}\cdot{2}}}{{{2}}}}+{\frac{{{2}\cdot{2}}}{{{2}}}}$$
Since the denominators are equal, combine the fractions: $$\displaystyle{\frac{{{a}}}{{{c}}}}\pm{\frac{{{b}}}{{{c}}}}={\frac{{{a}\pm{b}}}{{{c}}}}$$
$$\displaystyle={\frac{{{x}^{{{3}}}-{x}^{{{2}}}\cdot{2}+{2}\cdot{2}}}{{{2}}}}$$
Multiply the numbers: $$\displaystyle{2}\cdot{2}={4}$$
$$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{{3}}}-{2}{x}^{{{2}}}+{4}}}{{{2}}}}$$
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 $$\displaystyle\rightarrow{2}$$