Introduction

Chapter -2

Polynomials

Introduction

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,  are algebraic expressions. All 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 case, 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 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, the perimeter is given by 4x units. So, as the length of the side varies, the perimeter varies.
Expressions of this form are called polynomials in one variable. In the examples above, the variable is x. For instance, x³ – x² + 4x + 7is a polynomial in x. For instance, x³ – x² + 4x + 7 is a polynomial in x. Similarly, 3y² + 5y is a polynomial in the variable y and t² + 4 is a polynomial in the variable t.
In the polynomial x² + 2x, the expressions x² and 2x are called the terms of the polynomial. Similarly, the polynomial 3y² + 5y + 7 has there terms, namely, 3y², 5y and 7. Can you write the terms of the polynomial –x³ + 4x² + 7x – 2 ? This polynomial has 4 terms, namely, –x³, 4x², 7x and –2.
Each term of a polynomial has a coefficient. So, in –x³ + 4x² + 7x – 2, the coefficient of x³ is –1, the coefficient of x² is 4, the coefficient of x is 7 and –2 is the coefficient of x^o (Remember, x^o = 1). Do you know the coefficient of x in x² – x + 7? It is –1.
2 is also a polynomial. In fact, 2, –5, 7 etc. are example 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, look at the polynomial p(x) = 3x⁷   – 4x⁶  + x + 9. What is the term with the highest power of x ? It is 3x⁷. The exponent of x in this term is 7. Similarly, in the polynomial q(y) = 5y⁶ – 4y² – 6, the term with the highest power of y is 5y⁶ 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 3x⁷ – 4 x⁶   + x + 9 is 7 and the degree of the polynomial 5y⁶ – 4y² – 6 is 6. The degree of a non-zero constant  polynomial is zero.
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_nxⁿ + a_{n – 1}x^{x – 1} + … + a₁x + a_o

Where a_o, a₁, a₂, …, a_n are constants and a_n ≠ 0.

In particular, if a_{o ­}= a₁ = a₂ = a₃ = … = 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.