Polynomials Class 10 Notes

The Class 10 Chapter 2 notes are based on the CBSE board textbooks, covering every topic included in the syllabus. These notes are designed by experts who have written them in such a way that students can easily recall what they have read during their exams. These notes have tables, charts, and diagrams that will turn the learning process into a fun experience for students.

Often students need clarification about which study material to use while revising the syllabus. The textbooks are very lengthy, and it is difficult to revise concepts from them when the exams are near. Students have to refer to multiple books while studying since they need help understanding their textbooks. While revising, they have to make their own notes since the textbooks and reference books are far too lengthy. Making your own summary notes is a time-consuming activity.

To solve this problem, we have come up with Polynomial notes for Class 10 that are ready to use and effective. These concise and crisp notes will be the one-stop reference for students while studying for their exams. Students will not need to buy any extra reference books.

Algebraic Expression

An Algebraic expression consists of constants, variables, and algebraic operations. Algebraic expressions are made up of ‘terms’. A term is a product of one variable and one constant. For example, 3x is a term. It is a product of 3 (constant) and x (variable). Similarly, 16yx2   is a product of 16 (constant), x, and y (variables). Algebraic expressions are built using these terms. 

For instance, 3x+ 16+ yx2, the sum of the above terms, is an algebraic expression.

A polynomial is a type of algebraic expression where all variables have whole numbers as their exponents, and the constants can be any real numbers.

For example- 25x3 -4 +3y is a polynomial, as all the constants are real numbers and all the variables have whole numbers (2 and 1) as their exponents.

However, x2 +√x +3 is not a polynomial as the exponent of x (1/2) exponent is not a whole number.

Every polynomial has a degree. The degree is the highest exponent that a variable in the polynomial has. 

When we write the terms of a polynomial in the descending order of the exponents of their variables, it is said that the polynomial is written in its standard format. For example, if we are asked to write the polynomial u2 +25u3 +33 in the standard format, the answer will be 25u3 +u2 +33.

Types of Polynomials

The highest exponent of the polynomial is known as the degree of the polynomial. For example- The degree of u3 +25u +33 is 3 because 3 is the highest exponent of the variable u.

Polynomials are classified based on their degree. 

1. Linear Polynomial- A polynomial of degree 1 is called a linear polynomial. Some examples of linear polynomials are 2x +32, 86y+2x+3, and √3x+1.
2. Quadratic Polynomial- The term quadratic is derived from the term ‘square’ and refers to having an exponent 2. A polynomial with degree 2 is called a quadratic polynomial. For instance, 3x2 +5x +6/7, 23q+8q2 +3, and √5y2 are quadratic polynomials.
3. Cubic Polynomial- A polynomial of degree 3 is called a cubic polynomial. Some examples of cubic polynomials are 3x3 +4x +7, √7q+ 3y3+8, and t3. The most common form of a cubic polynomial is ax3 +bx2 +cx +d, where a is not equal to 0.

Zeroes of a Polynomial

Zeroes of a polynomial are the values that give an answer of 0 when substituted in place of the variables of the polynomial. Graphically speaking, these are the locations on the graph on which the value of the polynomial becomes 0. 

For a polynomial p(x), the value of x at which p(x) becomes equal to 0 is the zero of the polynomial.

Geometrical Meaning of the Zeroes of a Polynomial

Let us try understanding the geometrical meaning of the zeroes of linear and quadratic polynomials. For that, we will need to learn about their geometric representation first. 

Zeroes of a Linear Polynomial: The graph of a linear polynomial is a straight line. The line intersects the x-axis at one point. That point at which the graph cuts the x-axis is the zero of that polynomial. The coordinates of that particular point will reveal the zero of that polynomial. Since the point is on the x-axis, a linear polynomial has only one zero.

Zeroes of a Quadratic Polynomial: The graph of a quadratic polynomial is a parabola, i.e. it has a U shape. The U can be upwards or downwards depending on the value of the constant in that equation. If the value of the constant is positive, the U shape is upwards and if the value of the constant is negative, the parabola is downwards. Similar to linear equations, the value of zero is given by the point at which the graph intersects the x-axis. However, in the case of a quadratic equation, the graph can cut the x-axis at 1 or 2 points. In some cases, it will not cut the x-axis at all. As a result, a quadratic equation can have no zeroes or a maximum of 2 zeroes.

Zeroes of Other Polynomials: The shape of the graph of any other polynomial having a degree n depends on the value of n. As the value of n increases, the graph becomes steeper. It also moves towards the Y-axis as the value of n rises. If the value of n is even, the graph is seen in the first and the second quadrants. Similarly, if the value of n is odd, the graph is located in the first and third quadrants. 

Number of Zeroes of a Polynomial: The maximum number of zeroes a polynomial can have is equal to its degree. For example, a polynomial axn by c can have a maximum of n zeroes. The polynomial 3x3 +4x+6 will have a maximum of 3 zeroes.

Relationship of the Coefficients of a Polynomial with its Zeroes

In the case of linear polynomials, their zero is related to their coefficients. For example, if k is the zero of the polynomial ax+b, then k = -b/a, i.e. – (constant)/Coefficient of x.

Similarly, for quadratic or cubic polynomials, there is a relationship between the coefficients and the zeroes. 

For a quadratic polynomial ax2+bx+c, having 2 zeroes ∝ and ϒ, the product of the zeroes is equal to the value of the constant term divided by the coefficient of x2. Also, the sum of the two zeroes is equal to the negative value of the coefficient of x divided by the coefficient of x2. Therefore, we can conclude that, for a quadratic polynomial in the form ax2+bx+c, having two zeroes ∝ and ϒ,

 ∝+ ϒ = -b/a and

∝ ϒ = c/a.

In the case of a cubic polynomial ax3 +bx2 +cx+d having 3 zeroes  ∝ ,β and ϒ, 

 ∝+ β+ ϒ = -b/a,

∝β+ βϒ+ ∝ϒ=  c/a, and 

∝ β ϒ =  -d/a.

Division of Polynomials

There is a specific method used for dividing one polynomial by another polynomial. The method has 3 simple steps that you can follow. 

Step 1:  Arrange the terms of both the polynomials (dividend and divisor) in decreasing order of their degrees. This is the process of writing a polynomial in its standard form. For example, let’s assume that we want to divide 3x3 +x2 +2x +5 by x2 +2x+1.

Step 2:  Divide the term of the dividend, which has the highest degree ( 3x3), by the term having the highest degree in the divisor (x2). This gives the first term of the quotient, i.e. 3x. What remains is the new dividend, i.e., 5x2-x+ 5. 

Step 3: Divide the highest degree term of the new dividend -5x2-x +5 (i.e., -5x2) by the highest degree term of the divisor (x2). The answer will be -5. What remains is the new dividend, i.e. 9x+10.

Step 4: The degree of the remaining polynomial 9x+10 is 1. It is less than the degree of the divisor (2). Because of this, the division cannot be continued further. 

Division Algorithm for Polynomials

The division algorithm states that when we are dividing one polynomial by another, we get an equation:

Dividend = Divisor x Quotient +Remainder.

While dividing two polynomials p(x) and g(x), where g(x) is not equal to 0, we can find the q(x), i.e. Quotient, and r(x), i.e. the remainder, by using the division algorithm. The division algorithm is similar to Euclid’s division algorithm that we studied in the previous chapter. 

As per the algorithm,

p(x) = g(x) x q(x) +r(x),  where

p(x) = Polynomial 1,

g(x)= Polynomial 2,

q(x)= the Quotient, and

r(x)= the Remainder.

Let us check out an example of the division algorithm. We can verify the division algorithm by dividing 3x2-x3-3x+ 5 by x-1-x2

Step 1. First of all, we will rewrite the polynomials in their standard form, i.e., -x3 +3x2-3x +5 and -x2 +x-1. -x3+3x2-3x+5  is the dividend and -x2 +x-1 is the divisor. 

Step 2. By performing the process of dividing one polynomial by another, as described above, we will find out the quotient and remainder to verify this algorithm. The remainder is 3, and the quotient is x-2. 

Step 3: Now, we can verify the algorithm. The formula is:

Divisor x Quotient+ Remainder

= (-x2 +x-1) x (x-2)+ 3

= -x3 +x2-x+ 2x2– 2x+ 2+ 3

= -x3 +3x2-3x +5, which is the dividend from our example.

Benefits of Using Polynomials Revision Notes

• Revision notes can help students in revising the entire subject within a short period of time. Students do not have the time to read the entire textbook when the exam is just a few days away. Having concise, crisp, and reliable revision notes can help them revise all the important concepts quickly on the day before the exams.
• These revision notes are designed strictly as per the CBSE textbooks, and they cover all the topics mentioned in the textbook with examples and illustrations. Students need not worry about leaving any important topic while revising the subject before exams.
• Most students struggle with recalling what they have studied during the exams. The notes are written in simple language that is easy to remember. This will help the students understand the concepts and recall them during the exams.

Conclusion

Class 10 is an important year in the lives of students. The students have to study in a competitive manner for the first time. It is crucial that they study in a stress-free, relaxed manner. Having good study resources is the key to achieving this. Students need to have short study notes in their hands in order to study with a positive mindset.

Most students feel that Maths is a difficult subject to score. The Polynomial chapter feels very technical and difficult to understand. Class 10 Polynomial notes have been designed to make this chapter seem easy and manageable. These notes have been designed by subject matter experts who have studied the CBSE syllabus in great detail. They have worked hard to make these notes very simple and easy to understand.