MATHEMATICS CLASS IX
Jishnusagar V S S III B.Ed. Mathematics Karmela Rani Training College Kollam 1
NATIONAL ANTHEM Jana Gana Mana Adhinayaka Jay He Bharata Bhagya Vidhata Panjab Sindhu Gujarat Maratha Dravida Utkala Banga Vindhya Himachal Yamuna Ganga Uchchala Jaladhi Taranga Tava Subha Name Jaage Tave Subha Aashish Mange Gaahe Tava Jay Gaatha Jana Gana Mangal Daayak Jay He Bharat Bhagya Vidhata Jay he Jay he Jay he Jay Jay Jay Jay he
PLEDGE India is my country and all Indians are my brothers and sisters. I love my country and I am proud of its rich and varied heritage. I shall always strive to be worthy of it. I shall give respect to my parents, teachers and elders and treat everyone with courtesy.to my country and my people, I pledge my devotion. In their wellbeing and prosperity alone, lies my happiness.
PREAMBLE OF INDIAN CONSTITUTION
Dear students, Starting your day with a positive attitude towards
learning and your life of study can be an enormous help to make your day in school go smoothly. Just a few minutes of reflection and writing down one positive thought in a gratitude journal every day will gradually form a more positive outlook. You might not see the effects immediately but if you tend to worry, try it and see the difference it can make to your day. May your journey through this book make learning a joyful experience. With love and regard, Jishnusagar V S
Polynomials in one variable
Polynomials – Related Terminologies
Degree of a polynomial
What is a polynomial The word polynomial is derived from the Greek words poly(meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms". Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable.
1.2 POLYNOMIALS IN ONE VARIABLE Polynomials in one variable is an algebraic expression that consists of one variable in it. Some of the examples of polynomials in one variable are given below
𝑥² + 3𝑥 + 2
𝑦³ + 2𝑦² − 𝑦 + 5
Recall that a variable is denoted by a symbol that can take any real value. We use the letters 𝑥, 𝑦, 𝑧, etc. to denote variables. 2𝑥, – 𝑥, – 1/2 𝑥 are all algebraic expressions These expressions are of the form (a constant) × 𝑥. Now suppose we want to write an expression which is (a constant) × (a variable) and we do not know what the constant is. In such cases, we write the constant as 𝑎, 𝑏, 𝑐, etc. So, the expression will be 𝑎𝑥, say.
Fig 1.1 2
2 Consider a square of side 2 units (see Fig. 1.1). What is its perimeter? You know that the perimeter of a square is the sum of the lengths of its four sides. Here, each side is 2 units. So, its perimeter is 4 × 2, i.e., 8 units. What will be the perimeter if each side of the square is 10 units? The perimeter is 4 × 10, i.e., 40 units. In case the length of each side is x units (see Fig. 2.2), the perimeter is given by 4𝑥 units. So, as the length of the side varies, the perimeter varies. Fig 1.2
If you find the area of the square PQRS? It is 𝑥 × 𝑥 = 𝑥 2 square units. = 𝑥 2 is an algebraic expression. You are also familiar with other algebraic expressions like 2𝑥, 𝑥² + 2𝑥, 𝑥³– 𝑥² + 4𝑥 + 7. Note that, all the algebraic expressions we have considered so far have only whole numbers as the exponents of the variable. Expressions of this form are called polynomials in one variable. For instance, 2𝑥³ – 3𝑥² + 4𝑥 + 6 is a polynomial in x. Similarly, 𝑦² + 2𝑦 is a polynomial in the variable 𝑦 and 𝑚² + 4 is a polynomial in the variable 𝑚.
Consider the expression, 3𝑥 2 + √2𝑥 This has square root of 2 in it. But the operations on the variable x involve only squaring and multiplication by fixed numbers 3 and √2. So, this is a polynomial.
Consider another expression, 7 𝑥+ 𝑥 This involves the operation of taking reciprocal of the variable and so it is not a polynomial.
1.5 POLYNOMIALS – RELATED TERMINOLOGIES
The different terms related to polynomials are given below:
Terms In an expression, a term can be a variable or constant or a product of variable and constant. In the polynomial 𝑥³ + 2𝑥, the expressions 𝑥³ and 2𝑥 are called the terms of the polynomial Coefficient A coefficient is a numerical value, which is written along with a variable. So, in 8𝑥 3 − 4𝑥² + 5𝑥 – 3, the coefficient of 𝑥³ is 8, the coefficient of𝑥² is −4, the coefficient of 𝑥 is 5 and –3 is the coefficient of 𝑥 0 . Variable A variable is a letter that represents the unknown value in an expression. In 𝑥² + 2𝑥, the variable is 𝑥.
Constant A constant is a number, whose value never changes in an expression. Consider an example, 5𝑥 + 2 Here, The variable is 𝑥, coefficient is 5, constant is 2, and terms are 5𝑥 and 2.
If the variable in a polynomial is 𝑥, we may denote the polynomial by 𝑝(𝑥), or 𝑞(𝑥), or 𝑟(𝑥), etc. So, for example, we may write: 𝑝(𝑥 ) = 𝑥 3 + 5𝑥 – 3 𝑞(𝑥) = 3𝑥² – 1 𝑟(𝑦) = 𝑦³ + 4𝑦 + 2
1.4 DEGREE OF A POLYNOMIAL What is degree of a polynomial? The degree of a polynomial is the highest power of the variable in a polynomial expression.
For instance, 𝑝(𝑥 ) = 6𝑥³ + 2𝑥² + 4 is a polynomial. Here 6𝑥³, 2𝑥², 4 are the terms where 6𝑥³ is a leading term and 4 is a constant term. The coefficients of the polynomial are 6 and 2. The degree of the polynomial 6𝑥³ + 2𝑥² + 4 𝑖𝑠 3. Or, 6𝑥³ + 2𝑥² + 4 is a polynomial of degree 3. Similarly, for the polynomial 𝑝(𝑥 ) = 3𝑥 8 + 4𝑥 3 + 9𝑥 + 1 The degree is 8. The polynomial 3𝑥 8 + 4𝑥 3 + 9𝑥 + 1 is a polynomial of degree 8.
Rather than saying polynomial of degree 1, polynomial of degree 2, polynomial of degree 3 and so on, we can say first degree polynomial, second degree polynomial, third degree polynomial and so on. Based on the degree, we can write the general forms of all polynomials.
First degree polynomial Second degree polynomial Third degree polynomial
𝑎𝑥 + 𝑏 𝑎𝑥² + 𝑏𝑥 + 𝑐 𝑎𝑥³ + 𝑏𝑥 + 𝑐𝑥 + 𝑑
Here, the letters 𝑎, 𝑏, 𝑐, 𝑑 are coefficients.
ACTIVITY 1. Write the coefficients of 𝑥² in each of the following: (i) 3𝑥² + 𝑥 (ii) 12 – 𝑥² + 𝑥 (iii) 2 ³ + 2𝑥² (iv) √2𝑥 2 + 3𝑥 + 4
2. Which of the following are polynomials (i) 5x (ii) 0𝑥² + 2𝑥 + 4 (iii) √2𝑥 + 3𝑥 1 2 1 (iv) 𝑥 + 2 4𝑥 3.Write the degree of each of the following polynomials: (i) 𝑥³ + 8𝑥 + 7𝑥² (ii) 1 – 𝑦² (iii) 2𝑡 – 4 (iv) 9𝑦
4.Find 𝑝(1) and 𝑝(2) in the following polynomials, (i) 𝑝(𝑥) = 𝑥 + 5 (ii) 𝑝(𝑥) = 5𝑥 2 + 2 − 𝑥 (iii) 𝑝(𝑥) = 2𝑥 3 + 3𝑥 2 + 4𝑥 + 7 5.Write each of the operations below in an algebraic form and state whether it is a polynomial or not. (i) (ii) (iii) (iv)
Sum of a number and its square. Sum of a number and its reciprocal. Product of the sum and difference of a number and its square root. Sum of a number and its square root.