2.1 Introduction

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 :

and

$(x+y)^{2}=x^{2}+2 x y+y^{2}$

$(x-y)^{2}=x^{2}-2 x y+y^{2}$

$x^{2}-y^{2}=(x+y)(x-y)$

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.

2.2 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 $2 x, 3 x,-x,-\frac{1}{2} x$ are algebraic expressions. All these expressions are of the form (a constant) $\times x$. Now suppose we want to write an expression which is (a constant) $\times($ 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 $a x$, 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 $4 \times 3$, i.e., 12 units. What will be the perimeter if each side of the square is 10 units? The perimeter is $4 \times 10$, i.e., 40 units. In case the length of each side is $x$ units (see Fig. 2.2), the perimeter is given by $4 x$ units. So, as the length of the side varies, the perimeter varies.

Fig. 2.1

Fig. 2.2 polynomial in $x$. Similarly, $3 y^{2}+5 y$ is a polynomial in the variable $y$ and $t^{2}+4$ is a polynomial in the variable $t$.

In the polynomial $x^{2}+2 x$, the expressions $x^{2}$ and $2 x$ are called the terms of the polynomial. Similarly, the polynomial $3 y^{2}+5 y+7$ has three terms, namely, $3 y^{2}, 5 y$ and 7. Can you write the terms of the polynomial $-x^{3}+4 x^{2}+7 x-2$ ? This polynomial has 4 terms, namely, $-x^{3}, 4 x^{2}, 7 x$ and -2 .

Each term of a polynomial has a coefficient. So, in $-x^{3}+4 x^{2}+7 x-2$, 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 $x+\frac{1}{x}, \sqrt{x}+3$ and $\sqrt[3]{y}+y^{2}$. Do you know that you can write $x+\frac{1}{x}=x+x^{-1}$ ? 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 $x^{\frac{1}{2}}+3$. Here the exponent of $x$ is $\frac{1}{2}$, which is not a whole number. So, is $\sqrt{x}+3$ a polynomial? No, it is not. What about $\sqrt[3]{y}+y^{2}$ ? 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 :

$p(x)=2 x^{2}+5 x-3$

$q(x)=x^{3}-1$

$r(y)=y^{3}+y+1$

$s(u)=2-u-u^{2}+6 u^{5}$

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

Consider the polynomials $2 x, 2,5 x^{3},-5 x^{2}, y$ 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:

$p(x)=x+1$, $q(x)=x^{2}-x$, $r(y)=y^{9}+1$, $t(u)=u^{15}-u^{2}$

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

$p(x)=x+x^{2}+\pi$ $q(x)=\sqrt{2}+x-x^{2}$

$r(u)=u+u^{2}-2$

$t(y)=y^{4}+y+5$

Now, look at the polynomial $p(x)=3 x^{7}-4 x^{6}+x+9$. What is the term with the highest power of $x$ ? It is $3 x^{7}$. The exponent of $x$ in this term is 7. Similarly, in the polynomial $q(y)=5 y^{6}-4 y^{2}-6$, the term with the highest power of $y$ is $5 y^{6}$ 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 $3 x^{7}-4 x^{6}+x+9$ is 7 and the degree of the polynomial $5 y^{6}-4 y^{2}-6$ is 6 . The degree of a non-zero constant polynomial is zero.

Now observe the polynomials $p(x)=4 x+5, q(y)=2 y, r(t)=t+\sqrt{2}$ 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 $2 x-1, \sqrt{2} y+1,2-u$. 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 $a x+b$, where $a$ and $b$ are constants and $a \neq 0$ (why?). Similarly, $a y+b$ is a linear polynomial in $y$.

Now consider the polynomials :

$$ 2 x^{2}+5,5 x^{2}+3 x+\pi, x^{2} \text { and } x^{2}+\frac{2}{5} x $$

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 $5-y^{2}$, $4 y+5 y^{2}$ 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 $a x^{2}+b x+c$, where $a \neq 0$ and $a, b, c$ are constants. Similarly, quadratic polynomial in $y$ will be of the form $a y^{2}+b y+c$, provided $a \neq 0$ 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 $4 x^{3}, 2 x^{3}+1,5 x^{3}+x^{2}, 6 x^{3}-x, 6-x^{3}, 2 x^{3}+4 x^{2}+6 x+7$. 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 $a x^{3}+b x^{2}+c x+d$, where $a \neq 0$ 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

$$ a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} $$

where $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ are constants and $a_{n} \neq 0$.

In particular, if $a_{0}=a_{1}=a_{2}=a_{3}=\ldots=a_{n}=0$ (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, $x^{2}+y^{2}+x y z$ (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 later.

2.3 Zeroes of a Polynomial

Consider the polynomial $p(x)=5 x^{3}-2 x^{2}+3 x-2$.

If we replace $x$ by 1 everywhere in $p(x)$, we get

$p(1) =5 \times(1)^{3}-2 \times(1)^{2}+3 \times(1)-2$

$=5-2+3-2$

$=4$

So, we say that the value of $p(x)$ at $x=1$ is 4 .

Similarly,

$p(0) =5(0)^{3}-2(0)^{2}+3(0)-2$

$=-2$

Can you find $p(-1)$ ?

What is $p(1)$ ? Note that : $p(1)=1-1=0$.

As $p(1)=0$, we say that 1 is a zero of the polynomial $p(x)$.

Similarly, you can check that 2 is a zero of $q(x)$, where $q(x)=x-2$.

In general, we say that a zero of a polynomial $p(x)$ is a number $c$ such that $p(c)=0$.

You must have observed that the zero of the polynomial $x-1$ is obtained by equating it to 0 , i.e., $x-1=0$, which gives $x=1$. We say $p(x)=0$ is a polynomial equation and 1 is the root of the polynomial equation $p(x)=0$. So we say 1 is the zero of the polynomial $x-1$, or a root of the polynomial equation $x-1=0$.

Now, consider the constant polynomial 5 . Can you tell what its zero is? It has no zero because replacing $x$ by any number in $5 x^{0}$ still gives us 5 . In fact, a non-zero constant polynomial has no zero. What about the zeroes of the zero polynomial? By convention, every real number is a zero of the zero polynomial.

2.4 Factorisation of Polynomials

Let us now look at the situation of Example 10 above more closely. It tells us that since the remainder, $q\left(-\frac{1}{2}\right)=0,(2 t+1)$ is a factor of $q(t)$, i.e., $q(t)=(2 t+1) g(t)$ for some polynomial $g(t)$. This is a particular case of the following theorem.

Factor Theorem : If $p(x)$ is a polynomial of degree $n \geq 1$ and $a$ is any real number, then (i) $x-a$ is a factor of $p(x)$, if $p(a)=0$, and (ii) $p(a)=0$, if $x-a$ is a factor of $p(x)$.

Proof: By the Remainder Theorem, $p(x)=(x-a) q(x)+p(a)$.

(i) If $p(a)=0$, then $p(x)=(x-a) q(x)$, which shows that $x-a$ is a factor of $p(x)$.

(ii) Since $x-a$ is a factor of $p(x), p(x)=(x-a) g(x)$ for same polynomial $g(x)$. In this case, $p(a)=(a-a) g(a)=0$.

2.5 Algebraic Identities

From your earlier classes, you may recall that an algebraic identity is an algebraic equation that is true for all values of the variables occurring in it. You have studied the following algebraic identities in earlier classes:

Identity I : $(x+y)^{2}=x^{2}+2 x y+y^{2}$

Identity III : $(x-y)^{2}=x^{2}-2 x y+y^{2}$

Identity III : $x^{2}-y^{2}=(x+y)(x-y)$

Identity IV : $(x+a)(x+b)=x^{2}+(a+b) x+a b$

You must have also used some of these algebraic identities to factorise the algebraic expressions. You can also see their utility in computations.

Identity V : $(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x$

Remark : We call the right hand side expression the expanded form of the left hand side expression. Note that the expansion of $(x+y+z)^{2}$ consists of three square terms and three product terms.

Identity VI : $\quad(x+y)^{3}=x^{3}+y^{3}+3 x y(x+y)$

Also, by replacing $y$ by $-y$ in the Identity VI, we get

Identity VII : $(x-y)^{3}=x^{3}-y^{3}-3 x y(x-y)$

$$ =x^{3}-3 x^{2} y+3 x y^{2}-y^{3} $$

Identity VIII : $x^{3}+y^{3}+z^{3}-3 x y z=(x+y+z)\left(x^{2}+y^{2}+z^{2}-x y-y z-z x\right)$

Summary

In this chapter, you have studied the following points:

1. A polynomial $p(x)$ in one variable $x$ is an algebraic expression in $x$ of the form $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{2} x^{2}+a_{1} x+a_{0}$, where $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ are constants and $a_{n} \neq 0$.

$a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ are respectively the coefficients of $x^{0}, x, x^{2}, \ldots, x^{n}$, and $n$ is called the degree of the polynomial. Each of $a_{n} x^{n}, a_{n-1} x^{n-1}, \ldots, a_{0}$, with $a_{n} \neq 0$, is called a term of the polynomial $p(x)$.

2. A polynomial of one term is called a monomial.

3. A polynomial of two terms is called a binomial.

4. A polynomial of three terms is called a trinomial.

5. A polynomial of degree one is called a linear polynomial.

6. A polynomial of degree two is called a quadratic polynomial.

7. A polynomial of degree three is called a cubic polynomial.

8. A real number ’ $a$ ’ is a zero of a polynomial $p(x)$ if $p(a)=0$. In this case, $a$ is also called a root of the equation $p(x)=0$.

9. Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.

10. Factor Theorem : $x-a$ is a factor of the polynomial $p(x)$, if $p(a)=0$. Also, if $x-a$ is a factor of $p(x)$, then $p(a)=0$.

11. $(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x$

12. $(x+y)^{3}=x^{3}+y^{3}+3 x y(x+y)$

13. $(x-y)^{3}=x^{3}-y^{3}-3 x y(x-y)$

14. $x^{3}+y^{3}+z^{3}-3 x y z=(x+y+z)(x^{2}+y^{2}+z^{2}-x y-y z-z x)$