# 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.

Question
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.

2021-02-04
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 $$\displaystyle\rightarrow{2}$$

### Relevant Questions

Indicate whether the expression defines a polynomial function. $$\displaystyle{g{{\left({x}\right)}}}=−{4}{x}^{{5}}−{3}{x}^{{2}}+{x}−{2}$$ polynomial or not a polynomial If it is a polynomial function, identify the following. (If it is not a polynomial function, enter DNE for all three answers.) (a) Identify the leading coefficient. (b) Identify the constant term. (c) State the degree.
Indicate whether the expression defines a polynomial function. $$\displaystyle{P}{\left({x}\right)}=−{x}{2}+{3}{x}+{3}$$ polynomial or not a polynomial If it is a polynomial function, identify the following. (If it is not a polynomial function, enter DNE for all three answers.) (a) Identify the leading coefficient. (b) Identify the constant term. (c) State the degree.
Determine whether the following state-ments are true and give an explanation or counterexample.
a) All polynomials are rational functions, but not all rational functions are polynomials.
b) If f is a linear polynomial, then $$\displaystyle{f}\times{f}$$ is a quadratic polynomial.
c) If f and g are polynomials, then the degrees of $$\displaystyle{f}\times{g}$$ and $$\displaystyle{g}\times{f}$$ are equal.
d) To graph $$\displaystyle{g{{\left({x}\right)}}}={f{{\left({x}+{2}\right)}}}$$, shift the graph of f 2 units to the right.
1. Is the sequence $$0.3, 1.2, 2.1, 3, ...$$ arithmetic? If so find the common difference.
2. An arithmetic sequence has the first term $$a_{1} = -4$$ and common difference $$d = - \frac{4}{3}$$. What is the $$6^{th}$$ term?
3. Write a recursive formula for the arithmetic sequence $$-2, - \frac{7}{2}, -5, - \frac{13}{2} ...$$ and then find the $$22^{nd}$$ term.
4. Write an explicit formula for the arithmetic sequence $$15.6, 15, 14.4, 13.8, ...$$ and then find the $$32^{nd}$$ term.
5. Is the sequence $$- 2, - 1, - \frac{1}{2},- \frac{1}{4},...$$ geometric? If so find the common ratio. If not, explain why.
Write A if the sequence is arithmetic, G if it is geometric, H if it is harmonic, F if Fibonacci, and O if it is not one of the mentioned types. Show your Solution. a. $$\frac{1}{3}, \frac{2}{9}, \frac{3}{27}, \frac{4}{81}, ...$$ b. $$3, 8, 13, 18, ..., 48$$
a) Use base b = 10, precision k = 4, idealized, chopping floating-point arithmetic to show that fl(g(1.015)) is inaccurate, where $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{\frac{{1}}{{4}}}}-{1}}}{{{x}-{1}}}}$$ b) Derive the second order (n = 2) quadratic Taylor polynomial approximation for $$\displaystyle{f{{\left({x}\right)}}}={x}^{{\wedge}}\frac{{1}}{{4}},$$ expanded about a = 1, and use it to get an accurate approximation to g(x) in part (a). c) Verify that your approximation in (b) is more accurate.
Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$\displaystyle{P}{\left({x}\right)}={3}{x}^{{4}}-{5}{x}^{{3}}+{x}^{{2}}-{3}{x}+{5}{N}{S}{K}{Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}$$ Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial $$\displaystyle{R}{\left({x}\right)}={x}^{{5}}—{2}{x}^{{4}}+{3}{x}^{{3}}—{2}{x}^{{3}}+{3}{x}+{4}$$ in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$P(x) = 3x^{4} - 5x^{3} + x^{2} - 3x +5$$
$$Q(x) = (((3x - 5)x + 1)x 3)x + 5$$
The general term of a sequence is given $$a_{n} = (1/2)^{n}$$. Determine whether the sequence is arithmetic, geometric, or neither.