# Indicate whether the expression defines a polynomial function. P(x) = −x2 + 3x + 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.

Question
Polynomial arithmetic
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.

2021-02-21
Definition used: “A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication and non- negative integer exponents of variables. The polynomial equation is of the form $$\displaystyle{c}_{{n}}{x}^{{{n}}}+{c}_{{{n}-{1}}}{x}^{{{n}-{1}}}+\ldots+{c}_{{2}}{x}^{{2}}+{c}_{{1}}{x}+{c}_{{0}}$$ Here, c, is the leading coefficient, n is the highest degree and the constant is a) The given function is $$\displaystyle{P}{\left({x}\right)}=−{x}{2}+{3}{x}+{3}.$$ By the definition of polynomial, the above function is quadratic polynomial. The leading coefficient is –1. b) By the definition of polynomial equation, the constant term is 3. c) By the definition of polynomial equation, the degree is 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.
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.
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.
Consider the following sequence.
$$s_{n} = 2n − 1$$
(a) Find the first three terms of the sequence whose nth term is given.
$$s_{1} =$$
$$s_{2} =$$
$$s_{3} =$$
(b) Classify the sequence as arithmetic, geometric, both, or neither. arithmetic, geometric bothneither
If arithmetic, give d, if geometric, give r, if both, give d followed by r. (If both, enter your answers as a comma-separated list. If neither, enter NONE.)
Consider the following sequence. $$\displaystyle{s}_{{n}}={2}{n}−{1}$$ (a) Find the first three terms of the sequence whose nth term is given. $$\displaystyle{s}_{{1}}={N}{S}{K}{s}_{{2}}={N}{S}{K}{s}_{{3}}=$$ (b) Classify the sequence as arithmetic, geometric, both, or neither. arithmeticgeometric bothneither If arithmetic, give d, if geometric, give r, if both, give d followed by r. (If both, enter your answers as a comma-separated list. If neither, enter NONE.)
For the following exercise, for each polynomial $$\displaystyle{f{{\left({x}\right)}}}=\ {\frac{{{1}}}{{{2}}}}{x}^{{{2}}}\ -\ {1}$$: a) find the degree, b) find the zeros, if any, c) find the y-intercept(s), if any, d) use the leading coefficient to determine the graph’s end behavior, e) determine algebraically whether the polynomial is even, odd, or neither.
For the following polynomial, P(x) = x^6 – 2x^2 – 3x^7 + 7, find: 1) The degree of the polynomial, 2) The leading term of the polynomial, 3) The leading coefficient of the polynomial.
Given the following function: $$\displaystyle{f{{\left({x}\right)}}}={1.01}{e}^{{{4}{x}}}-{4.62}{e}^{{{3}{x}}}-{3.11}{e}^{{{2}{x}}}+{12.2}{e}^{{{x}}}-{1.99}$$ a)Use three-digit rounding frithmetic, the assumption that $$\displaystyle{e}^{{{1.53}}}={4.62}$$, and the fact that $$\displaystyle{e}^{{{n}{x}}}={\left({e}^{{{x}}}\right)}^{{{n}}}$$ to evaluate $$\displaystyle{f{{\left({1.53}\right)}}}$$ b)Redo the same calculation by first rewriting the equation using the polynomial factoring technique c)Calculate the percentage relative errors in both part a) and b) to the true result $$\displaystyle{f{{\left({1.53}\right)}}}=-{7.60787}$$
$$\displaystyle{f{{\left({x}\right)}}}={1.01}{e}^{{{4}{x}}}-{4.62}{e}^{{{3}{x}}}-{3.11}{e}^{{{2}{x}}}+{12.2}{e}^{{{x}}}$$
a) Use three-digit rounding frithmetic, the assumption that $$\displaystyle{e}^{{{1.53}}}={4.62}$$, and the fact that $$\displaystyle{e}^{{{n}{x}}}={\left({e}^{{{x}}}\right)}^{{{n}}}$$ to evaluate $$\displaystyle{f{{\left({1.53}\right)}}}$$
c) Calculate the percentage relative errors in both part a) and b) to the true result $$\displaystyle{f{{\left({1.53}\right)}}}=-{7.60787}$$
For each of the following sequences: @ identify if the sequence is arithmetic, geometric or quadratic’. Justify your response. @ assuming the first item of each sequence is a1, give an expression for aj. (In other words, find a formula for the i-th term in the sequence). @ if the sequence is arithmetic or geometric, compute the sum of the first 10 terms in the sequence $$i 2,-12, 72, -432, 2592,...$$
$$ii 9, 18, 31, 48, 69, 94,...$$
$$iii 14, 11.5, 9, 6.5, 4, 1.5,...$$