Polynomials-Types, Graphs, Factor and Remainder Theorem
Indtroduction :
Algebra is the branch of mathematics which explain the relation among the constants and variables.
Constants and variables :
Generally in Algebra, two types of symbols are used constants and variables (literals)
Constant :
It is a symbol whose value always remains the same, whatever the situation be, it is represented by C (or) K
Ex : \(
\frac{7}
{2},\sqrt 3 ,\, - 10,e,\,\pi
\)
Variable :
It is a symbol whose value changes according the situation
Ex : \(
x,\,y,\,z\,,\,\,\,bx,\,\,c + y,\,\,\,9z\,\,e + c
\)
Algebraic Expression :
An algebraic expression is a collection of terms separated by + and - symbol.
Ex : \(
7x + 11y,\,\,\,\,10y - 3x,\,\,\,3x + by - dz
\)
The various parts of an Algebraic expression that are seperated by + (or) - sign are called terms.
Ex : Algebraic expression NO.of terms terms
1) \(
- 36x
\) 1 \(
- 36x
\)
2) \(
ax + 5y - dz
\) 3 ax, -5y and dz
3) \(
\frac{7}
{x} - \frac{6}
{y} - \frac{{yz}}
{9} + 7
\) 4 \(
\frac{7}
{x},\frac{{ - 6}}
{y},\frac{{ - yz}}
{9}\,\,\,and\,\,\,7
\)
i) Monomial :
Mono means one An Algebraic expression having only one term is called a monomial.
Ex : \(
8,\,\,7a,\,\, - 11xyz\,\,\,
\) etc
ii) Binomial :
“Bi” means two. An algebraic expression having two terms is called a binomial
Ex :\(
ax + by,\,\,\, - 1 + \frac{a}
{2},\,\,\,\,2x^2 + 4z
\) etc
iii) Trinomial :
“Tri” means three An algebraic expression having three terms is called a trinomial
iv) Multinomial :
An algebraic expression having more then 3 terms is called a multinomial
Ex : 3a - 4x + 6y + z
v) Factors and Coefficents :
Each combination of the constants and variables which form a term is called factor.
Coefficent :
Any factor of a term is called the coefficent of the remaining term
for Ex : i) In -5x2y, 5 is coefficent of -x2y, -5 is coefficent of x2y
Note :
\(
- 13 = \,\, - 13 \times 1\,\,\, - 13 \times x^0
\)
coefficent of x0 is (-13)