Introduction

INTRODUCTION

In Class IX, you have studied polynomials in one variable and their degrees. Recall that if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x). For example, 4x + 2 is a polynomial in the variable x of degree 1, 2y² – 3y + 4 is a polynomial in the variable y of degree 2, 5x³ – 4x² + x-\(\sqrt2\) is a polynomial in the variable x of degree 3 and 7u⁶ –  u⁴ + 4u² + u+ 8 is a polynomial in the variable u of degree 6. Expressions like  etc., are not polynomials.

A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3, etc., are all linear polynomials. Polynomials such as 2x + 5 – x², x³ + 1, etc., are not linear polynomials.

A polynomial of degree 2 is called a quadratic polynomial. The name ‘quadratic’ has been derived from the word ‘quadrate’, which means ‘square’.  are some examples of quadratic polynomials (whose coefficients are real numbers). More generally, any quadratic polynomial in x is of the form ax² + bx + c, where a, b, c are real numbers and a0. A polynomial of degree 3 is called a cubic polynomial. Some examples of a cubic polynomial are . In fact,the most general form of a cubic polynomial is ax³ + bx² + cx + d, where, a, b, c, d are real numbers and a0.

Now consider the polynomial p(x) = x² – 3x – 4. Then, putting x = 2 in the polynomial, we get p(2) = 2² – 3 × 2 – 4 = – 6. The value ‘– 6’, obtained by replacing x by 2 in x² – 3x – 4, is the value of x² – 3x – 4 at x = 2. Similarly, p(0) is the value of p(x) at x = 0, which is – 4.

If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).

What is the value of p(x) = x² –3x – 4 at x = –1?

We have :

p(–1) = (–1)² –{3 × (–1)} – 4 = 0

Also, note that  p(4) = 4² – (3  4) – 4 = 0.

As p(–1) = 0 and p(4) = 0, –1 and 4 are called the zeroes of the quadratic polynomial x² – 3x – 4. More generally, a real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

You have already studied in Class IX, how to find the zeroes of a linear polynomial. For example, if k is a zero of p(x) = 2x + 3, then p(k) = 0 gives us 2k + 3 = 0, i.e., k =
In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e., k = .

So, the zero of the linear polynomial ax + b is .

Thus, the zero of a linear polynomial is related to its coefficients. Does this happen in the case of other polynomials too? For example, are the zeroes of a quadratic polynomial also related to its coefficients?

In this chapter, we will try to answer these questions. We will also study the division algorithm for polynomials.