Polynomials and Rational Expressions

Polynomials and Rational Expressions

(a) Types of Polynomials w.r.t. Terms

A polynomial having one term is called a monomial. e.g., 3, x, 0 etc.

A polynomial having two terms is called a binomial.

e.g. $\mathbf{a}+\mathbf{b},2\mathbf{x}+\mathbf{y},7\mathbf{a}\mathbf{b}-\frac{1}{3},{\mathbf{x}}^3+2\frac{1}{4}$$\mathbf{a} + \mathbf{b},2\mathbf{x} + \mathbf{y},7\mathbf{a}\mathbf{b} - \frac{1}{3},\mathbf{x}^{3} + 2\frac{1}{4}$ etc.

A polynomial having three terms is called a trinomial.

All other polynomials having more than three non-zero terms are called polynomials of four terms, five terms, six terms and so on.

When a polynomial is written in descending order, then the first coefficient is called the leading coefficient. e.g., in $\mathrm{p}(x)=3x^4-2x^2+1$$\mathrm{p}(x) = 3x^{4} - 2x^{2} + 1$, the leading coefficient is 3. Here $\mathrm{p}(x)$$\mathrm{p}(x)$ means that the variable in the polynomial is 'x'.

(b) Types of Polynomials w.r.t. Degree

Zero polynomial or no degree polynomial:

'0' is called a polynomial of no degree. Also, $0x^{3} + 0x$ is a no degree polynomial, because coefficients are always zero in zero polynomial.

Constant polynomial: A polynomial having degree zero is called a constant polynomial. e.g. 2, $-5,\frac{1}{2},\sqrt{5}$$-5,\frac{1}{2},\sqrt{5}$

​−5,21​,5

​ are all constant polynomials.

Linear polynomial: A polynomial having degree one is called a linear polynomial. e.g., $x,2x - y, -7xy^{0}$ etc.

Quadratic polynomial: A polynomial having degree two is called a quadratic polynomial. e.g., $2x^{2} + 7$, ax + 2xy + 3, $-\frac{3}{4}x{y}^0$$-\frac{3}{4}xy^{0}$ etc.

Cubic polynomial: A polynomial having degree three is called a cubic polynomial. e.g. $9x^{3} - 7x + 5$, $9xy,3x^{2}y-\frac{3}{4}z$$9xy,3x^{2}y - \frac{3}{4}z$ etc.

All other polynomials have no specific name w.r.t. degree but simply, we call them polynomials of degree four, degree five and so on.