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Chapter 2: Polynomials

    Introduction

    Geometrical Meaning of the Zeroes of a Polynomial

    Relationship between Zeroes and Coefficients of a Polynomial

    Exercise 2.1

    Exercise 2.2

    What have we discussed?

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Chapter 2: Polynomials > Introduction

Introduction

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 "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.

A polynomial is a mathematical expression that is made up of (such as x) and (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, 2y2−3y+4 is a polynomial in the variable y of degree 2, 5x3−4x2+x−2 is a polynomial in the variable x of degree 3 and 7u6−32u4+u−8 is a polynomial in the variable u of degree 6. Expressions like 1x−1, x+2,1x2+2x+3 etc., are not polynomials.

The degree of a polynomial is the power of the variable in the polynomial.

For example, in the polynomial 4x3+2x2−5x+3, the degree is 3 because the highest power of x is x3.

A linear polynomial has a degree of .

For example,2x – 3,3x+5,y+2 etc., are all linear polynomials. Polynomials such as 2x+5−x2, x3+1, etc., are not linear polynomials.

A quadratic polynomial has a degree of .

The name ‘quadratic’ has been derived from the word ‘quadrate’, which means .

2x2+3x−25,y2−2,2−x2+3, u3−2u2+5, 5v2−23v, 4z2+17 are some examples of quadratic polynomials (whose coefficients are real numbers). More generally, quadratic polynomial in x is of the form ax2+bx+c, where a, b, c are real numbers and a ≠ 0.

A cubic polynomial has a degree of .

Some examples of a cubic polynomial are 2−x3, x3, 2x3, 3−x2+x3, 3x3−2x2+x−1.

In fact, the most general form of a cubic polynomial is ax3+bx2+cx+d, where, a, b, c, d are real numbers and a ≠ 0.

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

Now consider the polynomial p(x) = x2−3x−4. Then, putting x = 2 in the polynomial, we get p(2) = 22−3×2−4 = .

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 = , which is .

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) = −12−3×−1−4 = .

Also, note that p(4) = 42−3x4−4 = .

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 previously, 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 =−32.

In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e., k=−ba.

So, the zero of the linear polynomial ax + b is =−ba= −Constant termCoefficient of x.

Thus, the zero of a linear polynomial is related to its .

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.