NCERT Class 9 Ganita Manjari Chapter 02 Introduction to Linear Polynomials PDF Download

Chapter 2: Introduction to Linear Polynomials

2.1 Introduction

We have learnt about algebraic expressions in earlier grades. In this chapter, we will learn about the special types of algebraic expressions called linear polynomials. Let us first consider a few examples of algebraic expressions.

Example 1: Raju went to a shop where there were sealed boxes of different colours on sale. The shop owner told him that the red boxes have 4 pens each and the blue boxes have 5 pencils each. Now, if Raju bought \( x \) red boxes and \( y \) blue boxes, how can he quickly figure out the total quantity of pens and pencils? Also, if he got 3 extra pens free, how many pens and pencils did he get altogether?

[Figure 2.1: Boxes of pens and pencils, See in your textbook]

Observe that \( x \) red boxes will have \( 4x \) pens and \( y \) blue boxes will have \( 5y \) pencils. Also, he got 3 extra pens free. Thus, the total number of pens and pencils is given by the algebraic expression \( 4x + 5y + 3 \). In this example, \( 4x \), \( 5y \) and \( 3 \) are terms of the expression, \( x \) and \( y \) are letter-numbers, the numbers \( 4 \) and \( 5 \) are the coefficients of \( x \) and \( y \), respectively, and \( 3 \) is a constant. From now onwards, we will use a widely used alternate word for letter-numbers: variables. Thus in the expression \( 4x + 5y + 3 \), we say that the variables used are \( x \) and \( y \).

Example 2: A rectangular garden of length \( l \) metres and width \( w \) metres has to be fenced and decorated. A wire fence is to be laid along the length costing Rs. 100 per metre and a wooden fence is to be built along the width costing Rs. 80 per metre. Special seeds have to be sown throughout the garden which will cost Rs. 50 per square metre.

[Figure 2.2: A rectangular garden with dimensions labelled, See in your textbook]

What will be the total cost incurred?

Cost of wire fencing along the garden length \( = 2l \times 100 = \text{Rs.} 200l \)

Cost of wooden fencing along the garden width \( = 2w \times 80 = \text{Rs.} 160w \)

Cost of sowing seeds throughout the entire garden (depends on the area) \( = 50 \times l \times w = \text{Rs.} 50lw \)

Total cost \( = \text{Rs.} (200l + 160w + 50lw) \).

Thus, \( 200l + 160w + 50lw \) is the algebraic expression for the total cost.

Think and Reflect

1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. How is it different from the algebraic expression in Example 1?

Teacher's Note

In Example 2, notice that the expression \( 200l + 160w + 50lw \) has a term with two variables multiplied together (\( lw \)). This makes it different from Example 1, where each term involves only one variable. When you solve problems, identify whether each term contains one variable or multiple variables - this is the key difference between a two-variable expression and other types.

Example 3: A wire of length 20 cm is bent in different ways to form rectangles. For example, we can have a rectangle with length 7 cm and width 3 cm. We can also have one of length 5.5 cm and width 4.5 cm. (Think of a few more ways of forming such rectangles.) Can you write an expression for the area of such rectangles?

If the length of the rectangle is \( x \) cm, then the width is \( (10 - x) \) cm. The expression for the area of these rectangles is \( x(10 - x) \) or \( 10x - x^2 \).

Think and Reflect

1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. Can you point out any similarity or difference between the algebraic expressions obtained in Examples 1 and 3?

Note that the algebraic expressions in Example 1 and Example 2 involve two variables, whereas the algebraic expression in Example 3 involves only one variable.

Expressions such as \( 4x \), \( x^2 + 1 \), \( 2y - 5 \), \( 5y^3 + y^2 + 2y - 1 \), \( 3z + 7 \) are algebraic expressions that involve only one variable: \( x \), \( y \) or \( z \).

In this chapter, we will restrict our discussion to algebraic expressions involving only one variable. You may have noticed that in an algebraic expression, the powers of a variable also appear. For example, in the expression \( x^2 + 5x + 1 \), the highest power of \( x \) is 2, whereas in the expression \( 5y^3 + y^2 - 8 \), the highest power of the variable \( y \) is 3. Further, in the expression \( 5y^3 + y^2 + 2y - 1 \), the coefficient of \( y^3 \) is 5, that of \( y^2 \) is 1, that of \( y \) is 2 and the constant term is \( -1 \). Such algebraic expressions involving one variable and its powers are called one-variable polynomials, univariate polynomials, or when the context is clear, simply polynomials. ('univariate' means 'having one variable'). The highest power of the variable in a polynomial is called its degree. For example:

1. \( 5y^3 + y^2 + 2y - 1 \) is a polynomial of degree 3. Such polynomials are called cubic polynomials.
2. \( x^2 + 5x + 1 \) is a polynomial of degree 2. Such polynomials are called quadratic polynomials.
3. \( 3z + 7 \) is a polynomial of degree 1. Such polynomials are called linear polynomials.
4. The constant 8 is a polynomial of degree 0 as it can be written as \( 8x^0 \) in which the power of the variable \( x \) is 0. Such polynomials are called constant polynomials.

Teacher's Note

Do not confuse the degree of a polynomial with the coefficient or the number of terms. The degree is only about the highest power of the variable that appears. For \( 5y^3 + y^2 + 2y - 1 \), even though there are four terms and several coefficients, the degree is 3 because the highest power is 3. In exams, always identify the highest power first, then you have found the degree.

Exercise Set 2.1

1. Find the degrees of the following polynomials:
   1. \( 2x^2 - 5x + 3 \)
   2. \( y^3 + 2y - 1 \)
   3. \( -9 \)
   4. \( 4z - 3 \)
2. Write polynomials of degrees 1, 2 and 3.
3. What are the coefficients of \( x^2 \) and \( x^3 \) in the polynomial \( x^4 - 3x^3 + 6x^2 - 2x + 7 \)?

Key Points

• A polynomial is an algebraic expression with one variable and its powers, like \( 3z + 7 \) or \( x^2 + 5x + 1 \).
• The degree of a polynomial is the highest power of the variable in it. For example, \( x^2 + 5x + 1 \) has degree 2.
• Linear polynomials have degree 1, quadratic polynomials have degree 2, and cubic polynomials have degree 3.
• In any term of a polynomial, the coefficient is the number multiplying the variable, and a term with no variable is called a constant term.

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