# 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 â€“ x

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

Â

x |
0 |
7 |

y |
7 |
0 |

Â

The graph of x + y = 7 intersect the x â€“ 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 x^{2} + 5x+ 6 = 0

Â

x |
-3 |
-2 |
-1 |
0 |
1 |
2 |

y = x |
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 ax^{2} + bx + c , a â‰ 0 the graph of the equation y = ax^{2} + 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 â€“2 and â€“3 are the zeroes of the quadratic polynomial. â€“2 and â€“3 are x-coordinates of the points where the graph of y = x^{2} + 5x + 6 intersect the x-axis. Thus the zeroes of the quadratic polynomial x^{2} + 5x + 6 are x â€“ coordinate where the graph of the above polynomial intersect the x-axis.

Hence for any quadratic polynomial the zeroes of a quadratic polynomial ax^{2} + bx + c â‰ 0 are precisely the x-coordinates of the points where the parabola representing y = ax^{2} + bx + c intersects the x â€“ axis.

Case (i)

Here it may cut the x-axis at 2 distinct points A and Aâ€™. We know that, the x-co-ordinate of A and Aâ€™ are two zeroes of the quadrate polynomial ax^{2} + bx + c in this case.

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**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 Aâ€™ in case (i) coincided here has become one point A.

The x-coordinate of A is the only zero for the quadratic polynomial ax^{2}+bx+c.

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**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)**

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So the quadratic polynomial ax^{2} + 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 x^{3} â€“ x. Let us list a few values of y corresponding to a few values for x.

Â

x |
-2 |
-1 |
0 |
1 |
2 |

y = 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 = x^{3} â€“ 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 â€“ coordinates of the only points where the graph of y = 4x^{3} â€“ 2x^{2} 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.