 # Determine whether g(x) = 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. vazelinahS 2021-02-03 Answered
Determine whether $g\left(x\right)=\frac{{x}^{3}}{2}-{x}^{2}+2$ 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.
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Step 1
Given:
$g\left(x\right)=\frac{{x}^{3}}{2}-{x}^{2}+2$
Step 2
Convert element to fracnion: ${x}^{2}=\frac{{x}^{2}2}{2},2=\frac{2×2}{2}$
$=\frac{{x}^{3}}{2}-\frac{{x}^{2}×2}{2}+\frac{2×2}{2}$
Since the denominators are equal, combine the fractions: $\frac{a}{c}±\frac{b}{c}=\frac{a±b}{c}$
$=\frac{{x}^{3}-{x}^{2}×2+2×2}{2}$
Multiply the numbers: $2×2=4$
$g\left(x\right)=\frac{{x}^{3}-2{x}^{2}+4}{2}$
Hence, it is a polynomial with its degree 3,
Step 3
Now, convert 3 degree polynomial into standard form which is
$a{x}^{3}+b{x}^{3}+cx+d$
Standard form $\to g\left(x\right)=\frac{{x}^{3}}{2}-2{x}^{2}+0x+2$
Leading term $\to \frac{1}{2}$
Constant term $\to 2$

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