Polynomials Class 10

If you are a class 10th student then you must have heard the term Polynomial many times in previous academic years. Probably, you would also know how to solve questions based on Polynomials. But Polynomial class 10 chapter is much more concrete and difficult in terms of theory as well as numerical. The concepts mentioned here are also used across Physics in the chapters voltage, energy and speed. Read the following blog to have a better understanding of this chapter!

What are Polynomials?

Polynomials are algebraic expressions with terms. It is also known as equations with multiple terms or parts. For example, 5×2+4y-12. These terms can be arranged or presented using addition, multiplication, subtraction or division. Most polynomials, as the name suggests, have more than one term. Polynomials can also have only one term, that is one constant, like 4, or 6, for example. However, they cannot have an infinite number of terms. 

Must Read: Score amazing marks with these Maths Tricks!    

What Are Polynomials Made Of?

Most polynomials are usually made up of 3 components or terms. These are usually variables, constants and exponents.

For example, continuing with the previous equation, 5×2+4y-12.

In this case, the variables are x and y.

The constants are 5,4 and 12.

The exponent is 2 in y2. Exponents in polynomials are only allowed to be 0,1,2,3 etc. Thus, terms with exponents like y-2 will not be considered polynomials, since the exponent is negative. These are a few more examples of polynomials: 4y, y-3, -5abc2 + 8/13 and 6.

In Polynomials class 10 chapter, you will find a series of questions where you will be determining the types, coefficients and degree of a polynomial. Wondering how to do that? Move on to the next section and find out!

Also Read: CBSE Class 10 Maths Syllabus

How Can Polynomials Be Combined?

Polynomials class 10 are usually combined using the functions of addition, subtraction, multiplication and division. However, keep in mind that a constant can never be divided by a variable. For example, 2/x will not be valid. Neither will the square root of x, for instance, because the exponent in square roots, translate to ½. Thus the square root of x is x^1/2. Since exponents in polynomials can only be 0,1,2,3 etc., this will not be valid. The square root of a constant, however, is valid as a polynomial.

Degree Of A Polynomial

The degree of a polynomial is the highest value of an exponent in a polynomial. For example, if a polynomial only consists of one term, 7, the degree would be 0, since there is no variable and no exponent. However, if a polynomial is x^2+3, the degree would be 2, since the highest exponent in this example, is 2, in x^2. This can be better demonstrated through a table, as given below.

Example of Polynomial	Degree	Polynomial Name
4	0	Zero or Constant Polynomial
4x+6	1	Linear Polynomial
4x2+6	2	Quadratic Polynomial
4x3+5x2+x+7	3	Cubic Polynomial
4x4+3x3+6x2+x+8	4	Quartic Polynomial

Terms Of A Polynomial

The components of polynomials, which are divided by addition and subtraction signs, are known as the terms of the polynomial. For example, 4x^2+5x-3 is a polynomial with three terms, which are 4x^2, 5x and 3. The number of terms decides what type of polynomial it is.

Types of Polynomials

 Based on the number of terms, class 10 Polynomials chapter classifies them into 3 kinds. These three kinds are monomials, binomials and trinomials.

Monomials

Monomials are polynomials which only have one term. This can be just a constant, such as 6, or a coefficient or constant attached to a variable as well, such as 6a. For a polynomial to be classified as a monomial, the term should be a term which is not equal to zero. Below are some examples of monomials- 4x, 5, 7xy, 3x^2.

Binomials

Binomials are polynomials which contain two terms, not more than that. Binomials can also be looked at, or broken down into two monomials, which are being either added or subtracted. Binomials can be made up of a constant and a variable, or even only two variables. Some examples of binomials are 5x + 4, 3xy – 6, ab^2 + ab, 3 + 4xy.

Trinomials

Trinomials, as the name suggests, is a polynomial consisting of only three terms and no more. These are the most common types of polynomials and can be used in a wide variety of problems, expressions and situations. Below are some examples of trinomials- 3×2+ 5x-4, -6x^2+3x-2, 5y+ 3x+4 and ab^2-8ab+7. 

Solved Examples of Polynomials

To have a better understanding of class 10 Polynomials chapter, have a look at the examples given below: 

In general, every polynomial will have this form: 

axⁿ + bx^n-1 + cx^n-2+ constant= ?

There are primarily 4 types of polynomials, zero polynomial, linear polynomial, quadratic polynomial and cubic polynomial, which are the most commonly used. Although there is a fifth type, quartic polynomial, it is hardly every used, and much beyond the scope of class 10. Hence, below are examples for zero, linear, quadratic and cubic polynomials.

 

Zero Polynomial

A zero polynomial occurs when the degree of x, that is n, is equal to zero. Since anything to the power of 0 is 1, you are then left with only the constant. Examples of zero polynomials: 

1. 6x^0, , where n, which is the power, is equal to 0. Since anything to the power of 0 is 1, when solved, this becomes 6 multiplied by 1, which is 6.  Hence, when solved, the answer is 6, which is constant, or coefficient of x^0.
2. 125x⁰= 125, since x⁰ is 1, so 125 multiplied by 1 is 125. 

Linear Polynomials

Linear polynomials are those where the highest power of x is equal to 1. Hence x¹ is x. These will always be in the form of ax+/- constant, where a is the coefficient of x. If there is no a in the formula, the coefficient of x is taken to be 1. Examples of linear polynomials: 

1. 6x-12= 0
   Thus, 6x= 12
   Thus, x = 12/6
   Thus x= 2
   Answer: x=2
2. 4x+24= 0
   Thus 4x= -24
   Thus x= -24/4
   Thus x= -6
   Answer: x= -6

Quadratic Polynomial

A quadratic polynomial is one in which the highest power of x is 2.  These will usually have 3 terms, as shown below. Examples of quadratic polynomials: 

a) 5x²+6x+1=0. We are using the below formula for finding out the roots
x=-bb2-4ac2a
Where a=5, b=6, c=1, referring to the original equation, 5x²+6x+1=0
After substituting the values in the formula, we get
x=-662-4*5*12*5
X= (-6+4)/10 and X=(-6-4)/10
X= -0.2, -1Answer= x= -0.2, -1

b) x² + 7x + 10 = 0 
By applying the same formula,
x=-bb2-4ac2a
x=-7-72-4*1*102*1
Answer= -2, -5

Also Read: Class 10 ICSE Syllabus

Cubic Polynomial

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A cubic polynomial is one in which the highest power of x is 3.  These will usually have 4 terms, as shown below. To solve these, you have to take either only x, or x and its coefficient as common examples of cubic polynomials:

a) x³-6x²+11x-6=0
(x-1)(x-2)(x-3)=0
x=1, x=2, x=3

b) 2x³-5x²-23x-10=0
(x+2)(2x²-9x-5)=0
(x+2)(2x+1)(x-5)=0
x=-3, x= -0.5, x= 5

c) 6x3−5x2−17x+6=0
(x−2)(6x2+7x−3)=0
(x−2)(2x+3)(3x−1)=0`
Answer x=2 or x=-(1/3) or x= 3/2

Polynomials can be of more than three terms as well for more complex problems. Yet, these three are the most common polynomials in class 10. With these three types, you can solve a wide range of problems and equations, and apply them to several situations. This was all on Polynomials class 10! If you want study notes, NCERT solutions or any other information regarding any class 10 topics then do checkout Leverage Edu blogs.