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Polynomials and Factorisation

    Introduction

    Polynomials in One Variable

    Degree of the Polynomial

    (a) Zeroes of a Polynomial

    (b) Zero of the Linear Equation in One Variable

    Dividing Polynomials

    Factorising a Polynomial

    Algebraic Identities

    Exercise 2.1

    Exercise 2.2

    Exercise 2.3

    Exercise 2.4

    Exercise 2.5

    What We Have Discussed

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Polynomials and Factorisation > Degree of the Polynomial

Degree of the Polynomial

Now, look at the polynomial px = 3x7−4x6+x+9.

What is the term with the highest power of x? It is . The exponent of x in this term is .

Similarly, in the polynomial q(y)= 5y6−4y2−6, the term with the highest power of y is and the exponent of y in this term is .

We call the highest power of the variable in a polynomial as the degree of the polynomial. So, the degree of the polynomial 3x7-4x6+ x + 9 is 7 and the degree of the polynomial 5y6 -4y2-6. The degree of a non-zero constant polynomial is zero.

Example

Find the degree of each of the polynomials given below:

Instructions

(i)x5-x4 + 3
The highest power of the variable is . So, the degree of the polynomial is .
(ii) 2- y2- y3 + 2y8
The highest power of the variable is . So, the degree of the polynomial is .
(iii) 2
The only term here is 2 which can be written as .So the exponent of x is . Therefore, the degree of the polynomial is .

Now observe the polynomials p(x) = 4x+5, q(y) = 2y, r(t) = t + 2 and s(u) = 3−u. Do you see anything common among all of them?

The degree of each of these polynomials is .

A polynomial of degree one is called a polynomial.

Some more linear polynomials in one variable are 2x−1, 2y+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 ax + b, where a and b are constants and a ≠ 0.

Similarly, ay + b is a linear polynomial in y.

Now consider the polynomials : 2x2 + 5, 5x2 + 3x , x2 and x2 + 25 x.

Do you agree that they are all of degree two? A polynomial of degree two is called a polynomial.

If you observe any quadratic polynomial in x then it is of the form ax2+ bx + c ,where a ≠ 0 and a, b, c are constants.

Similarly, quadratic polynomial in y will be of the form ay2+ by + c, provided a ≠ 0 and a, b, c are constants.

We call a polynomial of degree : a cubic polynomial. Some examples of a cubic polynomial in x are 4x3,2x3+1,5x3+x2,6x3-x,6-x3,2x3+4x2+6x+7

How many terms do you think a cubic polynomial in one variable can have?

It can have at most terms. These may be written in the form ax3+ bx2 + cx + d, where a ≠ 0 and a, b, c and d are constants.

Instructions

Instructions

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:

anxn + an−1xn−1 + ......... + a1x+ a0 = 0

In particular, if a0= a1= a2= a3= . . . = an= 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 .

So far we have dealt with polynomials in one variable only. We can also have polynomials in more than one variable. For example, x2+ y2 + xyz (where variables are x, y and z) is a polynomial in three variables. Similarly p2 + q10 + r (where the variables are p, q and r), u3 + v2 (where the variables are u and v) are polynomials in three and two variables, respectively. You will be studying such polynomials in detail later.