CBSE Tutorials

POLYNOMIALS

An expression of the form where all are real numbers and n is a non-negative integer, is called a polynomial.

The degree of a polynomial in one variable is the greatest exponent of that variable.

are called the co-efficients of the polynomial .

an is called constant term.

DEGREE OF A POLYNOMIAL

The exponent of the term with the highest power in a polynomial is known as its degree.

and are polynomials of degree 3 and 2 respectively.

Thus, is a polynomial of degree n, if

On the basis of degree of a polynomial, we have following standard names for the polynomials.

A polynomial of degree 1 is called a linear polynomial. Example: etc.

A polynomial of degree 2 is called a quadratic polynomial. Example: , etc.

A polynomial of degree 3 is called a cubic polynomial. Example: , etc.

A polynomial of degree 4 is called a biquadratic polynomial. Example: .

VALUE OF A POLYNOMIAL

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

e.g. : Value of at will be

\    

ZEROS OF A POLYNOMIAL

A real number a is a zero of polynomial if = 0.

The zero of a linear polynomial is . i.e.

Geometrically zero of a polynomial is the point where the graph of the function cuts or touches x-axis.

When the graph of the polynomial does not meet the x-axis at all, the polynomial has no real zero.

SIGNS OF COEFFICIENTS OF A QUADRATIC POLYNOMIAL

The graphs of are given in figure. Identify the signs of a, b and c in each of the following:

         (i)    We observe that represents a parabola opening downwards. Therefore, a < 0. We observe that the turning point of the parabola is in first quadrant where

\     Þ Þ                     

Parabola cuts y-axis at Q. On y-axis, we have x = 0.        Putting x = 0 in we get

So, the coordinates of Q are (0, c). As Q lies on the positive direction of y-axis. Therefore, c > 0.

Hence,         and

(ii)    We find that represents a parabola opening upwards. Therefore, The turning point of the parabola is in fourth quadrant.

\        

        Parabola cuts y-axis at Q and y-axis. We have x = 0. Therefore, on putting x = 0 in we get

So, the coordinates of Q are (0, c). As Q lies on negative y-axis. Therefore, c < 0.

Hence, a > 0, b < 0 and c < 0.

(iii)    Clearly, represents a parabola opening upwards.

Therefore, a > 0. The turning point of the parabola lies on OX.

\        

        The parabola cuts y-axis at Q which lies on positive y-axis. Putting
x = 0 in we get y = c. So, the coordinates of Q are (0, c). Clearly, Q lies on OY.

\        c > 0.

Hence,        a > 0, b < 0, and c > 0.

(iv)    The parabola opens downwards. Therefore, a < 0.

The turning point of the parabola is on negative x-axis,

\            

        Parabola cuts y-axis at Q (0, c) which lies on negative y-axis. Therefore, c < 0.

Hence,     and

(v) We notice that the parabola opens upwards. Therefore, a > 0.

Turning point of the parabola lies in the first quadrant.

\    

        As Q (0, c) lies on positive y-axis. Therefore, c > 0.

Hence, and c > 0.

(vi)    Clearly,

lies in the fourth quadrant.

\     

As Q (0, c) lies on negative y-axis. Therefore,
c < 0.

Hence, and c < 0.

Question:

The graphs of are given below for some polynomial . Find the number of zeros of in each case.

Solution:

(i)    The polynomial represented in (i) has no zero because its graph does not intersect x-axis at any point.

(ii)    The polynomial represented in (ii) has one zero because its graph intersects x-axis at one point.

(iii)    The polynomial represented in (iii) has three zeros because its graph intersects x-axis at three points.

(iv)    The polynomial represented in (iv) has two zeros because its graph intersect x-axis at two points.

(v)    The polynomial represented in (v) has four zeros because its graph intersect x-axis at four points.

(vi)    The polynomial represented in (vi) has three zeros because its graph intersects x-axis at three points.