Polynomials definition, formula and types
Class -10 chapter – 2, Polynomials
(A) What is a quantity?
A quantity is a thing which can be measured. There are two types of quantities:–
(1) Constant quantity
(2) Variable quantity
(B) What is a constant quantity?
Those quantities are considered as constant quantities which having a specific value and do not change its value with the change in the mathematical situation. All types of numbers come under this type of quantity.
Example:- 2, 3.4, 7/9 etc
‘x0’ is sometimes considered as constant term as x0= 1.
(C) What is a variable quantity?
Those quantities are assumed as variable quantities which have a specific value for a particular condition but change its value as the situation changes. Variable quantities are represented by letters of English alphabet. Variable quantity depends upon numbers used in the equation.
In simple words, It (variable quantity) can also said to be dependent upon the constant quantities. Because under the category of constant quantity, all types of numbers comes.
Examples :- x, y, z, a, b, c, P, Q, R etc.
(D) What are coefficients of variables?
Coefficients of a particular element in a specific term of an expression, polynomial or an equation are nothing but numerical or algebraic multiplying/multiplicative factors of that specific element.
But in general, we are asked to calculate coefficient for a particular variable in a specific term of an expression, polynomial or an equation.
Example : In an equation 5x + 3 = 0, the coefficient of x is 5.
(E) How to calculate coefficient/method for finding coefficient/steps for finding coefficient?
We can easily calculate coefficient of any element in a specific term of an expression/polynomial/equation with help of some steps given below
(i) Take an element into the consideration, the coefficient of which needs to be calculated inside a term.
(ii) Multiply rest elements except the considered element with each other. The product of rest elements will be coefficient.
(F) What are terms in algebra?
A term in algebra is basically a single expression. There are 5 types of terms:-
(1) Term is a single constant quantity. Examples :- 1, 2, √5 etc.
(2) Term is a single variable quantity. Examples :- a, b, c, x, y, z etc.
(3) Term is a combination of two or more constant quantities where connecting operators will be either multiplication or division or both. Examples :- 2/7, 3×√5, 7/√5 etc.
(4) Term is a combination of two or more variable quantities where connecting operators will be either multiplication or division or both. Examples :- a×b, x/y, xyz etc.
(5) Term is a combination of variable and constant quantities where connecting operators will be either multiplication or division or both. Examples:- 2x, 5/y, xyz/3 etc.
(G) What is an expression?
In the algebraic branch of the mathematics, an expression is defined as a mathematical statement in which numbers, variables and as per situation appropriate arithmetic operators in between them are present.
When more than one term is used in an expression, the operation must be either addition or subtraction or both.
Examples:- (x+1), (5x+y), (5x+y+z+1) etc.
(H) What is an equation?
When two expressions are connected with help of the symbol of equality, then such representation is called an equation.
Examples:- x+1=0
(I) How (x=0) is an equation?
We know that single variable is considered as a term. And single term may be assumed as an expression. Therefore, x represents an expression on LHS (the left hand side).
In same way, as per the definition of term, a constant is also considered as a term. Hence ‘0’ is a term.
Hence, zero is representing another expression on RHS (the right hand side).
Now, since 1st expression x on LHS is connected with the 2nd expression (0) on RHS with the help of the symbol of the equality. Therefore, x=0 is an equation.
(J) What is the degree of expression/equation/polynomial?
Degree of an expression or an equation or a polynomial is basically defined by the highest power or exponent of variable in the given expression/equation/polynomial.
In case of a constant quantity, the degree will be zero.
Example :- Degree of x2 + y = 0 is2.
Because, in the given equation the highest power/exponent of the variable is 2.
(K) Easiest method for finding degree?
We can easily find the degree of any equation or expression or polynomial with the help of following steps.
(1) Find exponent of each variable in each term.
(2) If more than one variable is present in one term, then add all exponents in that specific term.
(3) Compare sum of exponents for all terms present in the expression or equation.
(4) The highest exponent of variable will be the degree of the given expression or equation.
(L) Degree of a constant term?
Degree of a constant term is zero.
Find the degree of 5.
We can write 5 as,
5 = 5×1 {Power of anything results 1}
= 5 × x0
As the highest power/exponent of the variable x is 0, the degree of given expression is 0.
(M) Definition of Polynomials?
Polynomials are basically an expression in which one or more than one term are present where more than one term are connected to each other by either addition or subtraction or both operators.
For an expression to be a polynomial it must follow a condition that exponent of any variable in any term should not be negative.
(N) Basis of Classification of polynomials?
Classification of polynomials is done on the two following basis:
(a) On the basis of number of terms,
(b) On the basis of the degree of the polynomials.
| On the basis of number of terms | On the basis of the degree of the polynomials |
| (1) Monomials | (i) Constant polynomial |
| (2) Binomials | (ii) Linear polynomial |
| (3) Trinomials | (iii) Quadratic polynomial |
| (4) Quadronomials | (iv) Cubic polynomial |
| (5) Pentonomials ………….So on | (v) Quartic polynomial ………….So on |
(O) On the basis of number of terms, definition of different types of polynomials:-
(1) Monomials:-
A monomial is that type of polynomial in which there is only one term. For an expression (having one term) to be a monomial, the only term should not be equal to zero.
Examples :– (a) 2,
(b) xy
(c) xy5 etc.
(2) Binomials:-
A binomial is that type of polynomial in which there are two terms. Terms present in the binomial should be connected to each other by either addition or subtraction operators.
Examples :– (a) 2 +x,
(b) x + y,
(c) x + y/9
(3) Trinomials:-
A trinomial is that type of polynomial in which three terms are present. In this type of polynomial, terms are connected to each other by the symbol of either addition or subtraction or both.
Examples :– (a) 2+x+y,
(b) x-y-z
(c) -5-x-y etc
(4) Quadronomials:-
A quadronomial is that type of the polynomial in which four terms are present. In this type of polynomial also, terms are connected to each other by the symbol of either addition or subtraction or both.
Examples :– (a) 2+x+y-z,
(b) 2+x-y-z/7
(P) On the basis of the degree of the polynomial, definition of different types of polynomials:-
(i) Constant Polynomials:-
A constant polynomial is that type of polynomial in which there is only one constant but no variable is present. The graph of a constant polynomial is a horizontal straight line parallel to the x-axis as the value of the constant polynomial remains the same irrespective of the change in the variable. Due to the lack of variable quantity, the degree of a constant polynomial is zero. This is the main reason, it is sometimes called zero polynomial. The general form of a constant polynomial is f(x) = c where c is a constant quantity.
Examples :- (a) f(x) = 5
(ii) Linear Polynomials:-
Linear polynomial is that type of a polynomial the degree of which is 1. In simple words, the highest exponent of the variable will be 1 in the case of linear polynomial. The graph of a linear polynomial is a straight line which intersects the x-axis at only one point. The general form of a linear polynomial is f(x) = ax+b where (a)should not be equal to zero.
Examples :- (a) f(x) = 5x+2
(iii) Quadratic Polynomials:-
Quadratic polynomial is that type of polynomial in which the value of the degree is 2. The graph of a quadratic polynomial is a parabola. The general form of a quadratic polynomial is f(x) = ax2+bx+c where a should not be equal to zero because if it will happen, the term ax2 will be zero and the quadratic polynomial will be reduced to a linear polynomial.
Examples :- (a) f(x) = 5x2+3x+2
(iv) Cubic Polynomials:-
Cubic polynomial is that type of polynomial the degree of which is 3. The general form of a cubic polynomial is f(x) = ax3+bx2+cx+d where a should not be equal to zero. Because, if it happens, the cubic polynomial will be changed into a quadratic polynomial.
Example:- f(y) = x3+2x2+3x+4
(v) Quartic Polynomials:-
That polynomial will be considered as a quartic polynomial the degree of which is 4. The general form of a quartic polynomial is f(x)=ax4+bx3+cx2+dx+e where a should not be equal to zero.
(Q) Zeroes of a polynomial:-
Zero of a polynomial p(x) is basically that value of variable [used in the polynomial p(x)] for which the final value of p(x) is 0. In general, number of zeroes in a specific polynomial p(x) is equal to the number of degree in the same polynomial.
| Types of polynomial | Number of degree | Number of zeroes |
| Zero/constant polynomial | 0 | 0 |
| Linear polynomial | 1 | 1 |
| Quadratic polynomial | 2 | 2 |
| Cubic polynomial | 3 | 3 |
| Quartic polynomial | 4 | 4 |
(R) How to find zeroes of a polynomial graphically / method of finding zeroes graphically
(i) The zeroes of a polynomial correspond to the number of times its curve intersects the x-axis.
(ii) If the curve will touch the x-axis, it will be assumed that the curve has cut the x-axis.
(S) How to understand variables in a polynomial?
(i) In a polynomial p(x), the number of variable is one and the specific variable is x.
Example:- p(x)= x+2.
(ii) In a polynomial p(y), the number of variable is one and the specific variable is y.
Example:- p(y)= y2+2y+1.
(iii) In a polynomial p(x,y), the number of variables are two and specific variables are x and y.
Example:- p(x,y)=x2 +2xy+y2.
(iv) In a polynomial p(x,y,z), the number of variables are three and specific variables are x,yand z.
Example:- p(x,y,z)= x+y+z
(T) Sum of zeroes of a quadratic polynomial:-
sum of zeroes = (-1)×[coefficient of x]/[coefficient of x2]
(U) Product of zeroes of a quadratic polynomial:-
Product of zeroes=[constant term]/[coefficient of x2]
(V) What is a ‘mathematical situation’ in mathematics:-
We will try to understand the term ‘mathematical condition’ with help of two scenarios explained below:
| 1st scenario | 2nd scenario |
| x + 3 = 6 | x + 4 = 6 |
| or, x = 6 – 3 | or, x = 6 – 4 |
| or, x = 3 | or, x = 2 |
(i) In the 1st scenario, 3 has been added to the variable quantity x and being made equal to 6. In this situation, the value of the variable x is 3 which is fixed for the 1st scenario.
(ii) But as we change the number (used in the equation), the condition changes.
(iii) In the 2nd scenario, 4 is added to the variable quantity x and being made equal to 6. In this situation, the value of the variable x is 2 which is fixed for the 2nd scenario.
(iv) In this way, we can say that the value of a variable quantity remains constant for a specific condition but its value changes with the change in the condition.
