Hello Students, This is the chapter of 9th class the Introduction to Linear Polynomials, where now you connect with me. I will discuss about introduction of this chapter because this is one of the most important chapter of your 9th class new ncert book and syllabus.
In this chapter , there are many important topic . That you will read later in this chapter .
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?
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 `100 per metre and a wooden fence is to be built
along the width costing `80 per metre. Special seeds have to be sown
throughout the garden which will cost `50 per square metre.
What will be the total cost incurred?
Cost of wire fencing along the garden length = 2l × 100 = `200l
Cost of wooden fencing along the garden width = 2w × 80 = `160w
Cost of sowing seeds throughout the entire garden (depends on
the area) = 50 × l × w = `50lw
Total cost = ` (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?
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 – x2
.
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, x2
+ 1, 2y – 5, 5y3
+ y2
+ 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 x2
+ 5x + 1, the highest power of x is 2,
whereas in the expression 5y3
+ y2
– 8, the highest power of the variable
y is 3. Further, in the expression 5y3 + y2
+ 2y – 1, the coefficient of y3
is 5, that of y2
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:
(i) 5y3
+ y2
+ 2y – 1 is a polynomial of degree 3. Such polynomials are
called cubic polynomials.
(ii) x2 + 5x + 1 is a polynomial of degree 2. Such polynomials are called
quadratic polynomials.
(iii) 3z + 7 is a polynomial of degree 1. Such polynomials are called
linear polynomials.
(iv) The constant 8 is a polynomial of degree 0 as it can be written as
8×0
in which the power of the variable x is 0. Such polynomials are
called constant polynomials.
Conclusion