Class 9 Polynomials in One Variable

Topic Covered

♦ Introduction
♦ Polynomials in One Variable


You have studied algebraic expressions, their addition, subtraction, multiplication and division in earlier classes.

You also have studied how to factorise some algebraic expressions. You may recall the algebraic identities :

` color {red} { (x + y)^2 = x^2 + 2xy + y^2 } `

`color {blue } { (x – y)^2 = x^2 – 2xy + y^2 } `

and ` color { green } { x^2 – y^2 = (x + y) (x – y) } `

and their use in factorisation. In this chapter, we shall start our study with a particular type of algebraic expression, called polynomial, and the terminology related to it.

We shall also study the Remainder Theorem and Factor Theorem and their use in the factorisation of polynomials. In addition to the above, we shall study some more algebraic identities and their use in factorisation and in evaluating some given expressions.

Polynomials in One Variable

Let us begin by recalling 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.

Notice that `2x, 3x, – x, -1/2 x` are algebraic expressions. All these expressions are of the form`text ( (a constant) ) × x`. Now suppose we want to write an expression which is`text ( (a constant) ) × text ( (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.

However, there is a difference between a letter denoting a constant and a letter denoting a variable.

The values of the constants remain the same throughout a particular situation, that is, the values of the constants do not change in a given problem, but the value of a variable can keep changing.

Now, consider a square of side 3 units (see Fig. 2.1). 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 3 units. So, its perimeter is `4 × 3`, i.e., 12 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.

Can 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, x^2 + 2x, x^3 – x^2 + 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. In the examples above, the variable is x.

For instance, `x^3 – x^2 + 4x + 7` is a polynomial in `x`. Similarly, `3y^2 + 5y` is a polynomial in the variable` y` and `t^2 + 4` is a polynomial in the variable `t`.

In the polynomial `x^2 + 2x`, the expressions `x^2` and `2x` are called the terms of the polynomial. Similarly, the polynomial `3y^2 + 5y + 7` has three terms, namely, `3y^2, 5y `and 7.

Can you write the terms of the polynomial `–x^3 + 4x^2 + 7x – 2 ?` This polynomial has
`4` terms, namely, `–x^3, 4x^2, 7x` and `–2`.

Each term of a polynomial has a coefficient. So, in `–x^3 + 4x^2 + 7x – 2`, the coefficient of `x^3` is `–1`, the coefficient of `x^2` is 4, the coefficient of `x` is `7` and `–2` is the coefficient of `x^0` (Remember, `x^0 = 1`). Do you know the coefficient of `x` in `x^2 – x + 7?` It is `–1`.

`2` is also a polynomial. In fact, `2, –5, 7`, etc. are examples of constant polynomials. The constant polynomial 0 is called the zero polynomial. This plays a very important role in the collection of all polynomials, as you will see in the higher classes.

Now, consider algebraic expressions such as `x + 1/x , sqrt (x ) + 3` and `root 3 (y) + y^2`. Do you know that you can write `x +1/x =x + x^(-1)` ?Here, the exponent of the second term, i.e., ` x^(-1) ` is `-1` , which is not a whole number. So, this algebraic expression is not a polynomial.

Again, `sqrt (x) +3` can be written as `x^(1/2) + 3` . Here the exponent of x is `1/2` , which is not a whole number. So, is `sqrt (x) +3` a polynomial? No, it is not. What about `root 3 (y) + y^2 ?` It is also not a polynomial (Why?).

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 :

` color { red} { p(x) = 2x^2 + 5x – 3 } `

` color {green } { q(x) = x^3 –1 } `

` color { blue } { r(y) = y^3 + y + 1 } `

` color { orange } { s(u) = 2 – u – u^2 + 6u^5 } `

A polynomial can have any (finite) number of terms. For instance, `x^150 + x^149 + ...+ x^2 + x + 1` is a polynomial with `151` terms.

Consider the polynomials` 2x, 2, 5x^3, –5x^2, y` and `u^4`. Do you see that each of these polynomials has only one term? Polynomials having only one term are called monomials (‘mono’ means ‘one’).

Now observe each of the following polynomials:

`p(x) = x + 1, q(x) = x^2 – x, r(y) = y^30 + 1, t(u) = u^43 – u^2`

How many terms are there in each of these? Each of these polynomials has only two terms. Polynomials having only two terms are called binomials (‘bi’ means ‘two’).

Similarly, polynomials having only three terms are called trinomials (‘tri’ means ‘three’). Some examples of trinomials are

` color { green } { p(x) = x + x^2 + π } `,

` color {red } { q(x) = sqrt 2 + x – x^2 } `,

` color { blue } { r(u) = u + u^2 – 2 } `,

` color { orange } { t(y) = y^4 + y + 5 } `.

Now, look at the polynomial` p(x) = 3x^7 – 4x^6 + x + 9`. What is the term with the highest power of `x` ? It is `3x^7`. The exponent of `x` in this term is `7`.

Similarly, in the polynomial `q(y) = 5y^6 – 4y^2 – 6`, the term with the highest power of `y` is `5y^6` and the exponent of y in this term is `6`. We call the highest power of the variable in a polynomial as the degree of the polynomial.

So, the degree of the polynomial `3x^7 – 4x^6 + x + 9` is `7` and the degree of the polynomial `5y^6 – 4y^2 – 6` is `6`. The degree of a non-zero constant polynomial is zero.

Q 3220445311

Find the degree of each of the polynomials given below:

(i) `x^5 – x^4 + 3`

(ii) `2 – y^2 – y^3 + 2y^8`

(iii) `2`
Class 9 Chapter 2 Example 1

(i) The highest power of the variable is `5`. So, the degree of the polynomial is `5`.

(ii) The highest power of the variable is `8`. So, the degree of the polynomial is `8`.

(iii) The only term here is `2` which can be written as `2x^0`. So the exponent of `x` is `0`.
Therefore, the degree of the polynomial is `0`.

Now observe the polynomials `p(x) = 4x + 5, q(y) = 2y, r(t) = t + sqrt 2` and
`s(u) = 3 – u`. Do you see anything common among all of them? The degree of each of
these polynomials is one. A polynomial of degree one is called a linear polynomial.
Some more linear polynomials in one variable are` 2x – 1, sqrt 2 y + 1, 2 – u`. Now, try and
find a linear polynomial in x with `3` terms? You would not be able to find it because a
linear polynomial in `x` can have at most two terms. So, any linear polynomial in `x` will
be of the form `ax + b`, where a and b are constants and `a ≠ 0` (why?). Similarly,
`ay + b` is a linear polynomial in `y`.

Now consider the polynomials :

`2x^2 + 5, 5x^2 + 3x + π, x^2` and `x^2 + 2/5 x`

Do you agree that they are all of degree two? A polynomial of degree two is called
a quadratic polynomial. Some examples of a quadratic polynomial are `5 – y^2`,
`4y + 5y^2` and `6 – y – y^2`. Can you write a quadratic polynomial in one variable with four
different terms? You will find that a quadratic polynomial in one variable will have at
most `3` terms. If you list a few more quadratic polynomials, you will find that any
quadratic polynomial in x is of the form `ax^2 + bx + c`, where `a ≠ 0` and `a, b, c` are
constants. Similarly, quadratic polynomial in `y` will be of the form `ay^2 + by + c`, provided
`a ≠ 0` and `a, b, c` are constants.

We call a polynomial of degree three a cubic polynomial. Some examples of a
cubic polynomial in `x` are `4x^3, 2x^3 + 1, 5x^3 + x^2, 6x^3 – x, 6 – x^3, 2x^3 + 4x^2 + 6x + 7`. How
many terms do you think a cubic polynomial in one variable can have? It can have at
most `4` terms. These may be written in the form `ax^3 + bx^2 + cx + d`, where `a ≠ 0` and
`a, b, c` and `d` are constants.

Now, that you have seen what a polynomial of degree 1, degree 2, or degree 3
looks like, can you write down a polynomial in one variable of degree n for any natural
number n? A polynomial in one variable x of degree n is an expression of the form

`a_n x^n + a_(n-1) x^(n-1) + ..... + a_1 x + a_0`

where `a_0, a_1, a_2, . . ., a_n` are constants and `a_n ≠ 0`.

In particular, if `a_0 = a_1 = a_2 = a_3 = . . . = a_n = 0` (all the constants are zero), we get
the zero polynomial, which is denoted by `0`. What is the degree of the zero polynomial?
The degree of the zero polynomial is not defined.

So far we have dealt with polynomials in one variable only. We can also have
polynomials in more than one variable. For example,` x^2 + y^2 + xyz` (where variables
are `x, y` and `z`) is a polynomial in three variables. Similarly `p^2 + q^10 + r` (where the
variables are `p, q` and `r`), `u^3 + v^2` (where the variables are `u` and `v` ) are polynomials in
three and two variables, respectively. You will be studying such polynomials in detail