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.