Basic concept of polynomial :
Here we are going to see some basic concepts of polynomial.
An Algebraic expression that contains only one term is called a monomial.
3a, -2x2, 5xy
An Algebraic expression that contains only two terms is called a binomial.
x + y, 4a + 3b, 3x2 - 5xy
An Algebraic expression that contains only three terms is called a trinomial.
x + y + z, 2a + 3b - 4, 2a + 3c - 5
An expression containing a finite number of terms with non-zero coefficient is called a polynomial. In other words, it is an expression containing a finite number of terms with the combination of variables, whole number exponents of variables and constants.
a + b + c + d
7xy, 3abc - 10
2x + 3y + 5z
3x5 + 4x4 - 3x3 7x2 - 5 x + 4
Degree of the Polynomial :
The monomials in the polynomial are called the terms. The highest power of the terms is the degree of the polynomial. The coefficient of the highest power of x in a polynomial is called the leading coefficient of the polynomial.
2x5 - x4 + 7x3 - 6x2 + 12x - 4 is a polynomial in x. Here we have six monomials.
Degree of the polynomial is 5.
The leading coefficient of the polynomial is 2.
In an algebraic expression, we may find the following three parts.
What is term ?
A single variable or a constant or a combination of these as a product or quotient forms a term.
Examples of terms :
5, -a, 3ab, 21/7, ........... etc
Terms can be added or subtracted to form an expression.
What is factor ?
Consider the expression 3ab – 5a. It has two terms 3ab and -5a. The term 3ab is a product of factors 3, a and b. The term -5a is a product of -5 and a. The coefficient of a variable is a factor or factors.
In the term 3ab;
(i) the coefficient of ab is 3 (ii) the coefficient of a is 3b
(iii) the coefficient of b is 3a.
In the term –5a the coefficient of a is –5
What is constant ?
A number which is not having any variable with it is known as constant.
Like terms (or) Similar terms :
Like terms are the terms which have the same variables with same exponent for each variable.
7x, 3x, - 4x
Unlike terms or Dissimilar terms:
Unlike terms are the terms which have same variables or different variables.
If they have same variables, the exponents will not be same.
9x², 5xy, - 4xy², y, 6
Example 1 :
The coefficient of x4 in -5x7 + (3/7)x4 + 3 x3 + 7x2 - 1 is
(A) -5 (B) -3 (C) 3/7 (D) 7
There are five terms in the given polynomial.
The value in front of the variable x4 is know as coefficient of x4.
So coefficient of x4 is 3/7. Hence option C is correct.
Example 2 :
The coefficient of xy2 in 7x2 - 14x2 y + 14xy2 - 5 is ______
The value in front of the variable xy2 is know as coefficient of xy2.
Hence coefficient of xy2 is 14.
Example 3 :
The power of the term x3 y2 z2 is _________
Power of x term is 3.
Power of y term is 2.
Power of z term is 2.
3 + 2 + 2 = 7
Hence the power of given term is 7.
Example 4 :
The degree of the polynomial x2 - 5x2y3 + 30x3y4- 576 xy is
The Power of 1st term is 2.
Power of 2nd term is 5.
Power of third term is 7.
Power of fourth term is 2.
Hence the degree of the given polynomial is 7.
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