Polynomials
Polynomial is a monomial or sum/difference of monomials.
Monomials are called terms of the polynomial.
Note, that binomial is also polynomial.
Examples of polynomial are:
- $$${5}+{x}+{{x}}^{{2}}$$$
- $$${2}{{x}}^{{2}}+{{y}}^{{3}}{x}+{z}{{y}}^{{3}}$$$
- $$${10}{x}{{y}}^{{3}}-{z}{{y}}^{{2}}{x}$$$
Examples of expressions, that are not polynomials:
- $$${x}+\frac{{1}}{{x}}+{3}$$$ (second term is not a monomial)
- $$$\frac{{{2}-{x}}}{{{x}+{2}}}$$$ (division is not allowed)
Degree of the polynomial is the largest number among degrees of its monomials.
Leading term is a monomial with the largest degree. Leading coeffcient is a coefficient of the leading term.
For example, in polynomial $$${4}{{x}}^{{4}}{{y}}^{{3}}-{9}{{z}}^{{8}}{{y}}^{{7}}+{3}{{x}}^{{2}}$$$ first term has degree $$${4}+{3}={7}$$$, second term has degree $$${8}+{7}={15}$$$ and third term has degree $$${2}$$$. The largest of numbers 7, 15 and 2 is 15.
Thus, degree of the $$${4}{{x}}^{{4}}{{y}}^{{3}}-{9}{{z}}^{{8}}{{y}}^{{7}}+{3}{{x}}^{{2}}$$$ is 15. Its leading term is $$$-{9}{{z}}^{{8}}{{y}}^{{7}}$$$ and leading coefficient is $$$-{9}$$$.
Polynomial in one variable is a polynomial, that contains only one variable.
In general, it can be written as $$${a}_{{n}}{{x}}^{{n}}+{a}_{{{n}-{1}}}{{x}}^{{{n}-{1}}}+\ldots+{a}_{{2}}{{x}}^{{2}}+{a}_{{1}}{x}+{a}_{{0}}$$$, where $$${n}$$$ is positive integer.
Using above definitions, we find, that degree of such polynomial is $$${n}$$$, leading term is $$${a}_{{n}}{{x}}^{{n}}$$$ and leading coefficient is $$${a}_{{n}}$$$.
Examples of polynomials in one variable:
- $$$-{2}{{y}}^{{5}}+{{y}}^{{4}}+{3}{{y}}^{{2}}$$$ (degree is 5, leading term is $$$-{2}{{y}}^{{5}}$$$, leading coefficient is $$$-{2}$$$)
- $$${1}+{{x}}^{{3}}+{{x}}^{{2}}-{2}{{x}}^{{2}}$$$ (degree is 3, leading term is $$${{x}}^{{3}}$$$, leading coefficient is $$${1}$$$)
Depending on the degree, polynomial in one variable has different names:
- zero degree: constant. For example, $$${7}$$$.
- 1st degree: linear. For example, $$${2}{x}+{3}$$$.
- 2nd degree: quadratic. For example, $$${{x}}^{{2}}-{2}{x}+{5}$$$.
- 3rd degree: cubic. For example, $$$-{4}{{x}}^{{3}}+{2}{{x}}^{{2}}-{5}$$$.
- 4th degree: quartic. For example, $$${3}{{x}}^{{4}}-{2}{{x}}^{{3}}+{{x}}^{{2}}+{x}+{7}$$$.
Exercise 1. Determine whether the following is a polynomial: $$${{x}}^{{3}}{y}+{3}{{y}}^{{2}}-{z}$$$?
Answer: yes.
Exercise 2. Determine whether the following is a binomial: $$$\sqrt{{{y}}}+{{x}}^{{2}}{y}$$$?
Answer: no.
Exercise 3. Determine whether the following is a binomial: $$${3}{{x}}^{{2}}+{2}{x}+{1}$$$?
Answer: yes.
Exercise 4. Find degree of the following polynomial: $$${{x}}^{{3}}+{2}{x}{{y}}^{{3}}+{{z}}^{{5}}$$$?
Answer: 5.
Exercise 5. Find degree of the following expression: $$${{x}}^{{3}}+{2}-{{x}}^{{7}}+{5}{{x}}^{{2}}$$$?
Answer: 7.