# Introduction

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Polynomials are an important mathematical concept that appear in many different areas of mathematics and science. They are used to model and analyze various phenomena, such as the motion of objects, the behavior of electrical circuits, and the distribution of data.

Understanding polynomials is important for many practical applications, such as constructing graphs, modeling data, and solving real-world problems. For example, you might use polynomials to model the relationship between the size of a loan and the interest rate, or to predict the trajectory of a baseball.

By converting a problem into polynomials "space", it makes it easy for us to model things without actually doing the real experiments. The language of polynomials allows you to model many kinds of situations that occur in the real world: from the trajectory of a football, to determining the speed, deceleration and acceleration of an object launched straight up into the atmosphere, to modeling behaviors of the economy of a country over time. Once you convert a problem to x's and y's, your imagination is your limit.

Let us now look at some definitions:

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A **polynomial** is a mathematical expression that is made up of variables (such as x) and constants (such as numbers), and is combined using only the operations of addition, subtraction, and multiplication. For example, 4x + 2 is a polynomial in the variable x of degree 1,

The **degree of a polynomial** is the highest power of the variable in the polynomial. For example, in the polynomial

A **linear polynomial** has a degree of 1. For example, 2x – 3,

A **quadratic polynomial** has a degree of 2. The name ‘quadratic’ has been derived from the word ‘quadrate’, which means ‘square’.

A **cubic polynomial** has a degree of 3. Some examples of a cubic polynomial are

The **zeros of a polynomial** are the values of the variable (such as x) that make the polynomial equal to 0.

Now consider the polynomial p(x) =

The value ‘– 6’, obtained by replacing x by 2 in

Similarly, p(0) is the value of p(x) at x =

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) =

Also, note that p(4) =

As p(–1) = 0 and p(4) = 0, –1 and 4 are called the **zeroes of the quadratic polynomial** **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.