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Geometrical Meaning of the Zeroes of a Polynomial

Any real number m is a zero of a polynomial p(x) if p(m) = 0.

Let us consider the geometric representation of linear and quadratic polynomials and the meaning of their zeroes.

Consider a linear polynomial

x + y = 7, y = 7 – x

The graph of x + y = 7 is a straight line passing through the points (0, 7), (7, 0)
 

x

0

7

y

7

0

 


The graph of x + y = 7 intersect the x – axis at (7, 0). Hence the zero of x + y = 7 is (7,0) as it intersects the x-axis at (7, 0).

Next, we look at the Geometrical meaning of zeroes of a quadratic polynomial.

Consider the quadratic polynomial x2 + 5x+ 6 = 0
 

x

-3

-2

-1

0

1

2

y = x2 + 5x + 6

0

0

2

6

12

20


We locate the points listed above on a graph paper and the drawn figure can be seen below.


We see that for any quadratic polynomial ax2 + bx + c , a ≠ 0 the graph of the equation y = ax2 + bx + c has one of the two shapes either open upward like or open downwards like depending on whether a > 0 or a < 0
These curves are called parabolas

In the above table –2 and –3 are the zeroes of the quadratic polynomial. –2 and –3 are x-coordinates of the points where the graph of y = x2 + 5x + 6 intersect the x-axis. Thus the zeroes of the quadratic polynomial x2 + 5x + 6 are x – coordinate where the graph of the above polynomial intersect the x-axis.

Hence for any quadratic polynomial the zeroes of a quadratic polynomial ax2 + bx + c ≠ 0 are precisely the x-coordinates of the points where the parabola representing y = ax2 + bx + c intersects the x – axis.

Case (i)
Here it may cut the x-axis at 2 distinct points A and A’. We know that, the x-co-ordinate of A and A’ are two zeroes of the quadrate polynomial ax2 + bx + c in this case.

 

Figure 2.3 (i)

Figure 2.3 (ii)

Case (ii)
Here the graph cuts the x-axis at exactly one point where the distinct points have coincided.

The two points A and A’ in case (i) coincided here has become one point A.

The x-coordinate of A is the only zero for the quadratic polynomial ax2+bx+c.
 

Figure 2.4 (i)

Figure 2.4 (ii)

Case (iii)
The graph is either completely above the x-axis or completely below the x-axis. Hence it does not cut the x-axis at any point.

Figure 2.5 (i)

Figure 2.5 (ii)

 

So the quadratic polynomial ax2 + bx + c has no zeroes in this case.

Geometrically we can see a quadratic polynomial can have either two distinct zeroes or two equal zeroes or no zero. This indicates that a quadratic polynomial has almost 2 zeroes.

Now look at the cubic polynomial x3 – x. Let us list a few values of y corresponding to a few values for x.
 

x

-2

-1

0

1

2

y = x3 – x

-6

0

0

0

6


Here -1, 1, 0 are the zeroes of the cubic polynomial. These co-ordinates are the x -coordinates of the three points where the graph of y = x3 – x intersects the x-axis.

Similarly, in the case of a cubic polynomial , the zeroes of this polynomial are 0 and .

These values are the x – coordinates of the only points where the graph of y = 4x3 – 2x2 intersects the x-axis.

Hence there are at the most 3 zeroes for any cubic polynomial (i.e.) any polynomial of degree 3 can have at the most 3 zeroes.

In general, a polynomial of degree n has at most n zeroes.





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