What is Zeros of Polynomial and its Geometrical Meaning?

What are zeros of a polynomial?

We have learnt earlier in the chapter Polynomial, its Terms, Degree and Types, the meaning of polynomials and its various types. This chapter is about what are zeros of polynomials, understand the zeros of polynomials graphically and finding the value of zeros of polynomial graphically for a linear polynomial, a cubic polynomial and a biquadratic polynomial.

We can represent a polynomial on a graph with a curve which intersects the axis at specific points. The values of the intersecting points on the x-axis are the values that make the polynomial equal to zero. These points on the x-axis which make a polynomial equal to zero are called zeros of a polynomial.

Below is an example of a polynomial which is drawn as a curve on a graph.

Example of graph of polynomial

In this graph, the curve of polynomial y = x + 4 intersects the x-axis at x = - 4 and y-axis at y = 4 or the intersecting points on the x-axis for x = -4 is (-4,0) and and on the y-axis for y = 4 is (0,4).
As said above in the introduction, the values of x that lie on x-axis i.e. x = -4 is the zero of the polynomial.
How?
We can check it by putting the value of x = - 4 in the polynomial y = x + 4, if y comes equal to 0 for x = -4, then the value of x is zero of the polynomial.
Substitute value x = - 4 in polynomial y = x + 4.
y = (-4) + 4
y = -4 + 4
y = 0
∴ x=-4 is zero of the polynomial.

Definition

A real number k is the zero of polynomial p(x) if p(k)= 0.
Let's see what it says. If p(x) is a polynomial in x and k is any real number, then the value obtained by replacing x by k in p(x) is called the value of p(x) at x = k and is denoted by p(k).
If the value of k in a polynomial can make the value of polynomial to zero, then k is called zeros of polynomial. We can learn it by example, how we can find out the zeros of a polynomial.
Zeros of a polynomial are determined by putting a different value of x, which can make the value of polynomial to 0.

Examples of zeros of polynomial

Find zeros of polynomial p(x) = x² - 5x + 6
p(x) = x² - 5x + 6
Put value x = 2
p(2) = (2)² - 5(2) + 6
= 4 - 10 + 6
= 10 - 10
= 0
p(2) = 0
∴ 2 is zero of polynomial p(x)
Put value x = 3
p(3) = (3)² -5(3) + 6
= 9 - 15 + 6
= 15 - 15
= 0
∴ 3 is also zero of polynomial p(x)
So, 2 and 3 are two zeros of polynomial p(x).

Geometrical meaning of zeros of polynomial

How the polynomial curves drawn on a graph with a specific curve have already been discussed above. The shape of the curve of a polynomial varies with its polynomial degree. Linear polynomials have curves of unique shape and are different from quadratic polynomials. A biquadratic polynomial also has its own unique shape. Let's learn what the shapes are and how to find zeros of polynomials for linear, quadratic, cubic and biquadratic polynomials.

Linear polynomial

The general form of a linear polynomial is ax + b, where a ≠ 0. The graph of a linear polynomial is a straight line and it intersects the x-axis at exactly one point. So, the linear polynomial has only one zero.

Example of graph of linear polynomial

Find zeros of a linear polynomial, y = 2x + 6
To find the zeros, put y = 0
2x + 6 = 0
x = -3
∴ -3 is zero of y = 2x + 6 linear polynomial

Quadratic polynomial

The general form of a quadratic polynomial is ax² + bx + c, where a ≠ 0. The graph of quadratic polynomial has two shapes, one is known as open upwards or in shape of ∪ and another is open downwards or in shape of downwards ∩. These curves are also called parabolas. Let's find zeros of a quadratic polynomial in an example as below.

Example of graph of quadratic polynomial

What are the zeros of a quadratic polynomial, y = x² + 2x - 3
To find zeros of a polynomial, let's start by putting the value of x and find the corresponding y.
Put x = 0
y = (0)² + 2(0) - 3
y = 0 + 0 - 3
y = 0 - 3
y = -3
Put x = -3
y = (-3)² + 2(-3) - 3
y = 9 - 6 - 3
y = 9 - 9
y = 0
Put x = 1
y = (1)² + 2(1) - 3
y = 1 + 2 - 3
y = 3 - 3
y = 0
The zeros of the quadratic polynomial x² + 2x - 3 will be the x coordinates of points where the graph y = x² + 2x - 3 intersects the x-axis.
Therefore, -3 and 1 are zeros of the polynomial x² + 2x - 3 as graph y = x² + 2x - 3 intersects the x-axis at -3 and 1.

Therefore, for the quadratic polynomial ax² + bx + c, a ≠ 0, zeros of polynomials are x-coordinates of points where the parabola y = ax² + bx + c intersects the x-axis.
There exists three different shapes of graphs for the polynomial y = ax² + bx + c.

1. Graph of a quadratic polynomial with distinct zeros

When the graph of a quadratic polynomial ax² + bx + c cuts the x-axis at two distinct points A and B. The x-coordinate of A and B are two zeros of the quadratic polynomial ax² + bx + c.

Example of graph of quadratic polynomial with distinct zeros

The following two graphs show the two quadratic polynomials y = x² + 5x + 6 and y = -x² + 5x - 6.

Graph of a quadratic polynomial with distinct zeros

Both the graphs cuts the x-axis at two distinct points.
y = x² + 5x + 6 cuts the x-axis at A(-2, 0) and B(-3, 0).
Similarly, y = -x² + 5x - 6 cuts the x-axis at P(2, 0) and Q(3, 0).
∴ -2 and -3 are the two distinct roots of y = x² + 5x + 6.
Also, 2 and 3 are the two distinct roots of y = -x² + 5x - 6.

2. Graph of a quadratic polynomial with coincident zeros

The graph of the polynomial ax² + bx + c cuts the x-axis at exactly one point i.e. at two coincident points. So the two points A and B coincide here to become one point A. In the graph of quadratic polynomial ax² + bx + c, the x-coordinate of A is 0.

Example of graph of quadratic polynomial with coincident roots

This graph shows two quadratic polynomials y = x² + 2x + 1 and y = -x² + 2x - 1.

Graph of a quadratic polynomial with coincident roots

Each graph touches the x-axis at a point.
y = x² + 2x + 1 has a coincident point on x-axis at A(-1, 0).
Similarly, y = -x² + 2x - 1 cuts the x-axis at P(1, 0).
∴ -1 is the root of y = x² + 2x + 1.
Also, 1 is the root of y = -x² + 2x - 1.

3. Graph of a quadratic polynomial with no zeros

Here the graph of quadratic polynomial ax² + bx + c is completely above x-axis and completely below x-axis. It does not cut the x-axis at any point. So, the quadratic polynomial has no zero in this case.

Example of graph of quadratic polynomial with no zeros

This graph shows two quadratic polynomials y = -x² + x - 2 and y = x² + x + 2.

Graph of a quadratic polynomial with no zeros

Both graphs do not cut the x-axis at any point.
y = -x² + x - 2 cuts the y-axis at a point A(0, 2).
Similarly, y = x² + x + 2 cuts the y-axis at P(0, -2).
Hence, no root exists for y = -x² + x - 2.
Also, y = x² + x + 2 has no roots.

Therefore, we can summarise from the above three cases, a quadratic polynomial can have either two distinct zeros or two equal zeros or no zero. In other words, a quadratic polynomial can have at most two zeros.

Cubic polynomial

The general form of a cubic polynomial is ax³ + bx² + cx + d, where a ≠ 0. The graph of a cubic polynomial intersects the x-axis. The coordinates of the points where the x-axis intersect are the zeros of the cubic polynomial.

Example to find zeros cubic polynomial on graph

Find zeros of cubic polynomial y = x³ - x
To find zeros of a polynomial, let's start by putting the value of x and find the corresponding y.
Put x = 1
y = (1)³ - (1)
y = 1 - 1
y = 0
Put x = 0
y=(0)³ - (0)
y = 0 - 0
y = 0
Put x = -1
y = (-1)³ - (-1)
y = -1 + 1
y = 0
Put x = -2
y = (-2)³ - (-2)
y = -(-8) + 2
y = -6
The zeros of cubic polynomial y = x³ - x will be the x coordinates of points where the graph y = x³ - x intersects the x-axis.
Here 1, 0 and -1 are zeros of cubic polynomials as these are points where the graph intersects the x-axis.

Biquadratic polynomial

The general form of a biquadratic polynomial is ax⁴ + bx³ + cx² + dx + c, where a ≠ 0. The graph of a biquadratic polynomial intersects the x-axis at points, the coordinates of these points are the only zeros of the biquadratic polynomial.

Example of graph of biquadratic polynomial

Find zeros of a biquadratic polynomial, y = x⁴ - 6x³ - 4x² + 54x - 45
To find zeros of the polynomial, let's start by putting the value of x and find the corresponding y.
Put x = -3
y = (-3)⁴ - 6(-3)³ - 4(-3)² + 54(-3) - 45
y = (81) - 6(-27) - 4(9) - 162 - 45
y = 81 + 162 - 36 -162 - 45
y = 0
Put x = 1
y = (1)⁴ - 6(1)³ - 4(1)² + 54(1) - 45
y = 1 - 6 - 4 + 54 - 45
y = 0
Put x = 3
y = (3)⁴ - 6(3)³ - 4(3)² + 54(3) - 45
y = 81 - 162 - 36 + 162 - 45
y = 0
Put x = 5
y = (5)⁴ - 6(5)³ - 4(5)² + 54(5) - 45
y = 625 - 750 - 100 + 270 - 45
y = 0
The zeros of cubic polynomial y = x⁴ - 6x³ - 4x² + 54x - 45 will be the x coordinates of points where the graph y = x⁴ - 6x³ - 4x² + 54x - 45 intersects the x-axis.
So, zeros of the biquadratic polynomial y = x⁴ - 6x³ - 4x² + 54x - 45 are -3, 1, 3 and 5.

Frequently Asked Questions

1) What are the zeros of a polynomial?

The value which makes the value of a polynomial equal to zero is called zeros of a polynomial. i.e. k is said to be zeros of polynomial, when a polynomial p(x) becomes equal to zero for the value of k. i.e. when we put x = k in p(x) and p(k) = 0.

2) What is the shape of a graph of a linear polynomial?

The graph of a linear polynomial is always a straight line.

3) What is the shape of the graph of a quadratic polynomial?

The graph of a quadratic polynomial ax² + bx + c is a parabola. It opens upward ∪ if a > 0 and opens downward ∩ if a < 0.

4) How many zeros of a polynomial are there of a polynomial with degree n?

The polynomial with degree n can have number of zeros equal to n or less than n. i.e. number of zeros ≤ n.

Solved Examples

1) Find the zeros of x² - 9.

Here p(x) = x² - 9
(x)² -(3)² using identity (a)² - (b)² = (a+b)(a-b)
(x+3)(x-3)
p(x) = 0
(x+3)(x-3) = 0
x+3 = 0 or x-3 = 0
x = -3 or x = 3
zeros of p(x) are -3, 3