Expressions or Algebraic expressions (Polynomial, Rational Expression, Irrational Expression) || Degree of a Polynomial || Classification of Polynomials (Monomial, Binomial, Trinomial, and Multinomial)

Algebraic expressions

There are three kinds of algebraic expressions.
i) Polynomial Expression / Polynomial
ii) Rational Expression
iii) Irrational Expression

i)    Polynomial    Poly+nomial

• Polynomial is the combination of two words Poly means "Many" and Nomial means "Terms".
• A polynomial is an algebraic expression where all of the exponents of the variables are whole integers.

OR

An expression of the type:

$P(x)$ $=a_{0}x^n$ $+a_{1} x^{n-1}$ $+a_{2} x^{n-2}$ $ + .... + a_{n-1} x$ $+a_{n}$    $1$

where $n$ is a positive integer or zero, $a_{n}\neq 0$ and the coefficients $a_{0}$, $a_{1}$, $a_{2}$,...$a_{n}$ are real numbers is called a polynomial of degree $n$ in one variable $x$.

a)    When $n=0$ and $a_{n}$ $\neq 0$

    $P(x)=a_{0}x^0$ $=a_{0}$    $($ as $x^0$ $=1)$

This gives $a_{0}$ (which is a constant) and hence becomes a polynomial of degree zero.

b)    If all the coefficients in a polynomial are zero

i.e. If

$P(x)$ $=0.x^n$ $+0.x^{n-1}$ $+0.x^{n-2}$ $+....+0.x$ $+0$

then $P(x)$ $=0$, a constant polynomial with no particular degree

If we put $n=1,2,3$ in $1$, then the corresponding polynomials are called linear polynomials, Quadratic polynomials, and Cubic Polynomials in one variable $x$ of degree $1$, $2$, and $3$ respectively and so on.

Example of Polynomial:

1. $9x^4$ $+2x^3$ $-3x$ $-5$, $ax^2$ $+bx$ $+c$, $2x$ $+4$ are polynomials in one variable $x$
2. $2xy$ $-x^2$ $-yz^2$, $2xy^2$ $-5x^2y$ $+2xz$, $3xy$ $-z$ are all polynomials in three variables $x$, $y$, and $z$.

Degree of a Polynomial

• The degree of a polynomial is the sum of the exponents of the variables in the term with the greatest degree is known as the degree of a polynomial.
• A linear polynomial is a polynomial whose greatest degree is just one.

E.g.

1. $2xy$ $+3xyz$ $-5xy^2$                    has degree   $1$ $+1$ $+1$ $=3$
2. $5x^2y^2$ $-2xz$ $+4$                      has degree   $2$ $+2$ $=4$
3. $3xyz^2$ $-2xy^2z^2$ $+4xy$          has degree    $1$ $+2$ $+2$ $=5$
4. $2x^2$ $-4xy$ $+8$                           has degree    $2$
5. $3x$ $+4y$ $+3$                                has degree    $1$

Rational Expression

Those algebraic expressions can be written in the form of $\frac{p(x)}{q(x)}$, $q(x)\neq 0$ $($ where $p(x)$ and $q(x)$ are polynomials $)$ is called a rational expression.

E.g.

$i)$ $\frac{x^2+1}{x}$, $x\neq 0$, and $\frac {2x^3+5x^2-7x+2}{9x^4+4x^3-2x+8}$ are called rational expressions in one variable $x$.

$ii)$  $\frac{2xy^2-2xy+2}{x^2-4xy+y^2}$, and $\frac{3xz-4x^2+5z^2}{3xz^2+3x-4}$ are rational expressions in two variables $(x,y)$ and $(x,y)$

$iii)$ $\frac{5x^2-3xyz+4}{3x-4y-4}$, and $\frac {3x^2-4y+5z-3xy+4}{4xyz-2y+4z^2-3x+7}$ are rational expressions in three variables $x$, $y$, and $z$.

Irrational Expression

Those algebraic expressions which can not be written in the form of $\frac {p(x)}{q(x)}$, $\neq 0$ where $p(x)$ and $q(x)$ are polynomials are called an irrational expression.

E.g.

$\sqrt{x}$, $3x^{2}$ $\sqrt{y}$, $\frac {1}{\sqrt{x+1}}$, $\frac {x+3x}{\sqrt{x^2+4}}$, $\sqrt{136}$ $y-5$ $\frac {x}{\sqrt{y}}$, $x^{3/2}$ $-x^{1/2}y^{5/2}$ $-5z^{3}$ are all irrational expressions.

Classification of Polynomials

a) The terms of the polynomials provide for classification.

1. Monomial

• A monomial is a kind of polynomial that has just one term.

e.g.

$3x^2$, $2$ $\sqrt{3}xy^2z$, $axyz^{2}$, $\frac{1}{2}$ $\sqrt{3}$ $x^{2}$ $y^{2}$ $z^{2}$ are all monomials

2. Binomial

• A binomial is made up of the two words bi, which means "two," and nomal, which means "terms."
• A binomial is a polynomial only with two terms known as binomial

e.g.

$3x$ $+4y$, $9x^2y$ $-4xyz$, $\frac{1}{2}$ $\sqrt{5}$ $xyz^2$ $-2xy$, $\left( x-y \right)^2-b^2$ are all binomials.

3. Trinomial

Trinomials are polynomials made up of three terms.

e.g.

$3x^2y$ $+\sqrt{2}xyz$ $+2\sqrt{3}y^2$, $x^2$ $+2y$ $+y^2$, $9xy$ $+4x^2y$ $-7xy^2$, $5x$ $+3y$ $-2z$ are all trinomials.

4. Multinomial

• A multinomial is an algebraic expression that has many terms.

$x$ $+y$, $x^2$ $+2xy$ $+y^2$, $x^3$ $+3xyz$ $+4x^2y^2z^2$ $-y^3$, $5x^2$ $+9y^2$ $+3z62$ $-x$ $-xy$, $\sqrt{2}y^2$ $-\frac {1}{2}$ $x^2y$ $-xyz$ $-x$ $+2xy$ $+y$ are all multinomials

b). The polynomial can be classified with respect to their degree

1. Linear Polynomial

• A linear polynomial is a polynomial whose maximum degree is just one.

e.g

$ax+b$ has degree "one" in one variable $x$.
$ax+by+c$ has degree one in two variables $x$, $y$ and $a$, $b$, $c$ are arbitrary constants.
$ax$ $+by$ $+cz$ $+d$ has degree one in three variables $x$, $y$, and $z$ and $a$, $b$, $c$, $d$ are arbitrary constants.

2. Quadratic Polynomial

• A polynomial whose highest degree is $2$ is called a quadratic polynomial.