Polynomials: 9 Math
What is Polynoimal?: 9th math
Solution of NCERT Exercise 2.1 class ninth math
Expressions having many terms are called POLYNOMIAL.
The word "Polynomial" comes from two words "Poly" and "Nomen".
"Poly" is a Greek word which means "many" and "Nomen" is a Latin word which means "Name or Term".
Thus, "Polynomial" = Poly + Nomial = Many + Term
Definition of Polynomial
An expression having variables and coefficients which involves the basic operations addition, subtraction and multiplication only and non-negative integer exponents of variables is called A POLYNOMIAL.
This means, an expression in which there are
(a) variables and coefficients
(b) consists of only three basic operation i.e. addition, multiplication and subtraction
(c) and non negative integer exponents of variables
is considered as Polynomial.
Example
2, 2x, 3x, 3x + 2, x2 + x + 7, etc. are some examples of polynomials
But there are some restrictions to be a Polynomial.
(1) Expression with division by a variable is not a polynomial
Example
`1/x`, `2/x-1, 1/(x-2)`, etc. are not polynomial.
(2) A polynomial cannot have a negative integer exponents of a variable
Example
Thu, x-2, `sqrt(x)`, 3x2 + x-3, etc. are not Polynomials.
But, `1/8x` is a Polynomial. Because division by a constant is allowed in a polynomial.
Constant and Variables of a Polynomial
Generally, x, y, z, etc. are considered as variable in a Polynomial. And 1, 2, 3, 4, . . . are considered as constant in a Polynomial.
Example
In a polynomial 2x + 3, 2 and 3 are constants and x is variable.
Coefficient
The multiplicative factor of a term in a polynomial is called Coefficient of that very term.
Example
3x2 + 2y + z +1
In this polynomial, the coefficient of x2 is equal to 3
And the coefficient of y = 2
And the coefficient of z = 1
Types of Polynomial
Polynomials are categorized on the basis of terms in a polynomial. These types are monomial, binomial, trinomial, quadrinomial, quintinomial, etc.
Monomial
Expression having only one term is called a MONOMIAL.
Example
2x, 2xy, 2x2, 2x2z, etc.
Since these expressions given in the example having only one term, thus these are considered as Monomials.
Binomial
Expression having two terms is called a BINOMIAL.
Example
(a) 2x + y
Since, in the given expression 2x + y, there are two terms, viz. 2x and y, thus, this is known as a Binomial.
(b) 3x2 + y is an example of binomial, since it has two terms 3x2 and y.
Other examples of Binomials
2x2 +3, 3x + 5y, 5z3 + 5, etc. are some examples of Binomial as these expressions have two terms.
Trinomial
An expression having three terms is called a TRINOMIAL.
Example
3x + y + 5 is an example of a trinomial, since it has three terms 3x, y and 5.
Similarly, 2x + 3y + z, 5x + 5 + y, 2x +9 + z, etc. are some more examples of trinomials.
Similarly, expressions having 4 terms are called QUADRINOMIAL. Expressions having 5 terms are called Quintinomial and so on.
Terms and Exponents
The power of a variable in a polynomial is called exponent. And number of expressions in the form of addition or subtraction is called terms.
In the given polynomial 2 is the power of variable x and 3 is the power of variable y. And there are three terms, 5x2, y3 and 2 in the given polynomial. Since this polynomial has total three terms, thus it is a trinomial.
Monomial and Polynomial
Expressions having only one term are known as Monomial and expressions having more than one term are known as Polynomial.
Thus, a binomial, a trinomial, a quadrinomial, etc. are called Polynomials.
Constant Polynomial
2, 3, -5, 7, etc. are known as CONSTANT POLYNOMILAS.
Zero Polynomial
0 is also a polynomial. And this is called Zero Polynomial.
If the variable in a polynomial is x, it can be denoted as p(x), or q(x) or r(x), etc. Thus,
p(x) = 4x2 + x + 3
Or, q(x) = x3 + x2 + 2
Or, r(x) = 5x + y, etc.
Degree of Polynomial
The highest power of the variable in a polynomial is denoted as the Degree of the Polynomial.
Example
(a) p(x) = 4x2 + x + 3
Thus, degree of this polynomial = 2. Because the highest power of variable x in this polynomial is 2.
If the degree of polynomial is 2, then it is called a Quadratic Polynomial
(b) q(x) = x3 + x2 + 2
Thus, degree of this polynomial = 3. Because the highest power of variable x in this polynomial is 3.
If the degree of polynomial is 3, then it is called a Cubic Polynomial
(c) r(x) = 5x + y, etc.
Thus, degree of this polynomial = 1. Because the highest power of variable x in this polynomial is 1.
If the degree of polynomial is 1, then it is called a Linear Polynomial
Degree of a non-zero polynomial
The degree of a non-zero polynomial is zero.
Solution of Polynomial NCERT Exercise 2.1 class ninth math
NCERT Exercise 2.1 Polynomials class ninth math Question (1) Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x2 - 3x + 7
Answer : This expression is polynomial in one variable.
Explanation
Since this expression has only one variable x, thus this is a polynomial in one variable.
(ii) `y^2 + sqrt2`
Answer: This expression is a polynomial in one variable.
Explanation
Since, this expression has only one variable y, and the exponent of the variable is a whole number, thus this is a polynomial in one variable.
(iii) `3sqrtt + tsqrt2`
Answer: This expression is not a polynomial.
Explanation
Since, in this expression exponent of variable `t` is `1/2`, which is not a whole number, and thus this is not a polynomial.
(iv) `y+2/y`
Answer: This expression is not a polynomial.
Explanation
Since, in this expression exponent of one of the variable `y` is `-1`, which is not a whole number, and thus this is not a polynomial.
(v) x10 + y3 + t50
Answer: This expression is not a polynomial in one variable. Rather this expression is a polynomial in three variables.
Explanation
Since, in this expression there are three variables x, y and t, and hence this is not a polynomial in one variable.
NCERT Exercise 2.1 Polynomials class ninth math Question (2) Write the coefficients of x2 in each of the following:
(i) 2 + x2 + x
Answer
The coefficient of x2 = 1
(ii) 2 – x2 + x3
Answer
The coefficient of x2 = –1
(iii) `pi/2x^2+x`
Answer
The coefficient of x2 = `pi/2`
(iv) `sqrt2 x-1`
Answer
The coefficient of x2 = 0
NCERT Exercise 2.1 Polynomials class ninth math Question (3) Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Answer
One of the binomial of degree 35 is x35 + 2
And one of the monomial of degree 100 is x100
Explanation
(a) An expression with two terms is called binomial. And a polynomial having highest exponent of variable equal to 35 is called a polynomial of degree 35.
(b) An expression with only one terms is called monomial. And a polynomial having highest exponent of variable equal to 100 is called a polynomial of degree 100.
NCERT Exercise 2.1 Polynomials class ninth math Question (4) Write the degree of each of the following polynomials:
(i) 5 x3 + 4 x2 + 7x
Answer
The degree of this polynomial = 3
Explanation
Because the highest value of exponent of variable is 3.
(ii) 4 – y2
Answer
The degree of given polynomial = 2
(iii) `5t –sqrt7`
Answer
The degree of polynomial = 1
(iv) 3
Answer
The degree of polynomial = 0
Explanation
Because 3 is a constant polynomial and the degree of a non-zero constant polynomial is zero (0).
NCERT Exercise 2.1 Polynomials class ninth math Question (5) Classify the following as linear, quadratic and cubic polynomials.
(i) x2 + x
Answer: Quadratic polynomial.
Explanation
Since, the degree of given polynomial = 2. And if the degree of a polynomial is equal to 2, then such polynomial is called a quadratic polynomial
(ii) x – x3
Answer: Cubic polynomial.
Explanation
Since, the degree of given polynomial = 3. And if the degree of a polynomial is equal to 3, then such polynomial is called a cubic polynomial
(iii) y + y2 + 4
Answer: Quadratic polynomial.
Explanation
Since, the degree of given polynomial = 2. And if the degree of a polynomial is equal to 2, then such polynomial is called a quadratic polynomial
(iv) 1 + x
Answer: Linear polynomial.
Explanation
Since, the degree of given polynomial = 1. And if the degree of a polynomial is equal to 1, then such polynomial is called a Linear polynomial
(v) 3t
Answer: Linear polynomial.
Explanation
Since, the degree of given polynomial = 1. And if the degree of a polynomial is equal to 1, then such polynomial is called a Linear polynomial
(vi) 3 t
Answer: Linear polynomial.
Explanation
Since, the degree of given polynomial = 1. And if the degree of a polynomial is equal to 1, then such polynomial is called a Linear polynomial
(vi) r2
Answer: Quadratic polynomial.
Explanation
Since, the degree of given polynomial = 2. And if the degree of a polynomial is equal to 2, then such polynomial is called a quadratic polynomial
(vii) 7 x3
Answer: Cubic polynomial.
Explanation
Since, the degree of given polynomial = 3. And if the degree of a polynomial is equal to 3, then such polynomial is called a cubic polynomial
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