# Geometrical Meaning of the Zeroes of a Polynomial

Complete above steps to enable this content

## Geometrical meaning of a zero of a linear polynomial

Why are the zeroes of a polynomial so important? To understand that, first lets look at the geometrical representation of linear and quadratic polynomials and the geomterical meaning of their zeroes.

When you just look at the equations with x and y, polynomials may appear daunting for some. But polynomials become beautiful when you represent them geometrically. Once you are able to visualize the polynomial equation on a graph you can understand the behaviour of the problem in a much better way.

For example, lets look at the equation y = 2x + 3. What does this mean geometrically? This looks very abstract. But if we substitute different values for x and get y we get multiple points in the co-ordinate system like:

x | -2 | 2 |
---|---|---|

y = 2x+3 | -1 | 7 |

When we plot these points in the co-ordinate system we get a graph like below. This makes it much easier to visualize the equation and we can imagine how the different values of x are going to affect y.

Complete above steps to enable this content

From the graph we can see that the graph of y = 2 x + 3 intersects the x - axis at

You also know that the zero of 2x + 3 is

The graph of y=2x+3 intersects the x-axis in

The linear polynomial ax + b, a ≠ 0, has exactly one zero, namely, the x-coordinate of the point where the graph of y = ax + b intersects the x-axis, which is the point (

Complete above steps to enable this content

## Geometrical meaning of a zero of a quadratic polynomial

Consider the quadratic polynomial

x | |
---|---|

-2 | |

-1 | |

0 | |

1 | |

2 | |

3 | |

4 | |

5 |

Complete above steps to enable this content

From the table above, the zeros of the quadratic polynomial

In fact, for any quadratic polynomial

For any quadratic polynomial, i.e., the zeroes of a quadratic polynomial

Complete above steps to enable this content

**From the above examples we observe that there are 3 cases, any quadratic equation will have 0 or 1 or 2 zeroes.**

**Case(i)**The x-coordinates are the **two zeroes** of the quadratic polynomial a

**Case(ii)**The x-coordinates is the **one zero** for the quadratic polynomial a

**Case(iii)**The x-coordinates is the **no zero** for the quadratic polynomial a

Graphs are given below. **Drop that into concerned boxes.**

So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also means that a polynomial of degree 2 has atmost two zeroes.

The zeros of a polynomial are important because they can provide valuable information about the behavior of the polynomial and its graph.

Zeros indeed play a fundamental role in understanding and interpreting the characteristics of polynomial functions. Here's a breakdown of how zeros contribute to polynomial analysis:

**Finding X-Intercepts:** Zeros of a polynomial p(𝑥) are the values of 𝑥 where p(𝑥)=0. These zeros correspond to the 𝑥 -intercepts of the graph of 𝑦=𝑝(𝑥), representing the points where the graph crosses the 𝑥 -axis. By setting p(x)=0 and solving for 𝑥, you can pinpoint these critical points on the graph.

**Understanding End Behavior:** The leading term of a polynomial (the term with the highest degree) dictates the end behavior of the polynomial's graph. For instance, in the polynomial p(x)= a

**· If 𝑛 (degree) is even and 𝑎>0, the graph approaches positive infinity as 𝑥 moves toward positive or negative infinity.**

**· If 𝑛 is even and 𝑎<0, the graph approaches negative infinity in the same manner.**

**· If 𝑛 is odd and 𝑎>0, the graph will rise to positive infinity when 𝑥 is positive and fall to negative infinity when 𝑥 is negative.**

Complete above steps to enable this content

## Geometrical meaning of a zero of a cubic polynomial

Lets look at some cubic polynomials and plot them to get their geometrical meaning. From that let's try to find some properties.

**(i) y = x3− 4x**

This clearly has

**(ii) y = x**

Move your mouse over the graph and find the zero polynomial of this graph. This has

**(iii) y = x**

Move your mouse over the graph and find the zero polynomial of this graph. This has

Complete above steps to enable this content

Note that 0 is the only zero of the polynomial

From the examples above, we see that there are at most 3 zeroes for any cubic polynomial. In other words, any polynomial of degree 3 can have at most three zeroes.

**Remark** : In general, given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.