Geometric Meaning Of The Zeroes Of A Polynomial
Let us consider linear polynomial ax + b. The graph of y = ax + b is a straight line.
For example : The graph of y = 3x + 4 is a straight line passing through (0, 4) and (2, 10).
(i) Let us consider the graph of y = 2x – 4 intersects the x-axis at x = 2. The zero 2x – 4 is 2. Thus, the zero of the polynomial 2x – 4 is the x-coordinate of the point where the graph y = 2x – 4 intersects the x-axis.
(ii) Let us consider the quadratic polynomial x2 – 4x + 3. The graph of x2 – 4x + 3 intersects the x-axis at the point (1, 0) and (3, 0). Zeroes of the polynomial x2 – 4x + 3 are the x-coordinates of the points of intersection of the graph with x-axis.
The shape of the graph of the quadratic polynomials is curve and the curve is known as parabola.
(iii) Now let us consider one more polynomial –x2 + 2x + 8. Graph of this polynomial intersects the x-axis at the points (4, 0), (–2, 0). Zeroes of the polynomial –x2 + 2x + 8 are the x-coordinates of the points at which the graph intersects the x-axis. The shape of the graph of the given quadratic polynomial is inverted curve and the curve is known as parabola.
Cubic polynomial: Let us find out geometrically how many zeroes a cubic has.
Let consider cubic polynomial x3 – 6x2 + 11x – 6.
Case 1: The graph of the cubic equation intersects the
x-axis at three points (1, 0), (2, 0) and (3, 0). Zeroes of the given polynomial are the
x-coordinates of the points of intersection with the x-axis.
Case 2: The cubic equation x3 – x2 intersects the x-axis at the point (0, 0) and (1, 0). Zero of a polynomial x3 – x2 are the x-coordinates of the point where the graph cuts the x-axis.
Zeroes of the cubic polynomial are 0 and 1.
Case 3: y = x3
Cubic polynomial has only one zero.
In brief : A cubic equation can have 1 or 2 or 3 zeroes or any polynomial of degree three can have at most three zeroes.
Remarks : In general, polynomial of degree n, the graph of y = p(x) passes x-axis at most at n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.
Example: Which of the following correspond to the graph to a linear or a quadratic polynomial and find the number of zeroes of polynomial.
Sol. (i) The graph is a straight line so the graph is of a linear polynomial. The number of zeroes is one as the graph intersects the x-axis at one point only.
(ii) The graph is a parabola. So, this is the graph of quadratic polynomial. The number of zeroes is zero as the graph does not intersect the x-axis.
(iii) Here the polynomial is quadratic as the graph is a parabola. The number of zeroes is one as the graph intersects the x-axis at one point only (two coincident points).
(iv) Here, the polynomial is quadratic as the graph is a parabola. The number of zeroes is two as the graph intersects the x-axis at two points.
(v) The polynomial is linear as the graph is straight line. The number of zeroes is zero as the graph does not intersect the x-axis.
(vi) The polynomial is quadratic as the graph is a parabola. The number of zeroes is 1 as the graph intersects the x-axis at one point (two coincident points) only.
(vii) The polynomial is quadratic as the graph is a parabola. The number of zeroes is zero, as the graph does not intersect the x-axis.