The value of variable for which the value of polynomial becomes zero is known as the Zero of the polynomial.
Geometrical Meaning of the Zeroes of a Polynomial
A polynomial can be represented geometrically through a coordinate plane. The variable is represented along xaxis and the polynomial is represented along yaxis. The ordered pairs of their corresponding values are then plotted and joined to obtain the graph of the polynomial.
Geometrically, zeroes of a polynomial means points where graph intersect with xaxis i.e., where y or p(x) is zero.
Graph of a linear polynomial
General form of a linear polynomial
ax + b
Let p(x) = 2x +1 be any linear polynomial.
To plot the graph we need different values of the x and p(x) corresponding to each other,
x

1

2

3

p(x)

3

5

7

We can observe that the graph of the polynomial is a straight line.
The coordinate of the point Q is (−1/2 , 0) which means the value of p(x) is 0 at x = −1/2 . Thus, the zero of the p(x) is −1/2 .
Hence, for a linear polynomial ax + b, a ≠ 0, the graph of p(x) = ax + b is a straight line which intersects the xaxis at exactly one point, (−b/a , 0).
∴ The linear polynomial ax + b, a ≠ 0, has exactly one zero, the xcoordinate of the point where the graph of p(x) = ax + b intersects the xaxis.
Graph of a Quadratic polynomial
General form of a quadratic polynomial
ax^{2} + bx + c
Let p(x) = 2x^{2} + x −1 be any quadratic polynomial.
x

−2

−1

0

1

2

p(x)

5

0

−1

2

9

We can observe that the graph of the polynomial is a curve called parabola.
The curve intersects xaxis at two points (1,0) and (1/2 ,0), therefore, zeroes of p(x) are 1 and 1/2 .
For a > 0, parabola open upwards or open downwards if a < 0
.
For a quadratic polynomial ax^{2} + bx + c, a ≠ 0, the graph is always a parabola but it’s position and orientation can be different.
Case I ↬
The graph cuts xaxis at two distinct points Q’ and Q.
The xcoordinates of Q’ and Q are the two zeroes of the quadratic polynomial, ax^{2} + bx + c in this case.
Case II ↬
The graph cuts xaxis at exactly one point, i.e., two points Q’ and Q of Case (i) coincide here to become one point Q.
The xcoordinate of Q is the only zero for the quadratic polynomial, ax^{2} + bx + c in this case.
Case III ↬
The graph is either completely above the xaxis or completely below the xaxis. So, it does not cut the xaxis at any point.
So, the quadratic polynomial, ax^{2 }+ bx + c has no zero in this case.
∴ We can observe geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero.
Thus, a polynomial of degree 2 has at most two zeroes.
Graph of a Cubic polynomial
General form of a cubic polynomial
ax^{3} + bx^{2} + cx + d
Let p(x) = x^{3} − 4x be any cubic polynomial.
x

−2

−1

0

1

2

p(x)

0

3

0

−3

0

The curve intersects xaxis at three points (2,0), (0,0) and (2,0) therefore, zeroes of p(x) =x^{3} − 4x are −2, 0 and 2.
Let p(x) =x^{3} be any cubic polynomial.
x

−2

−1

0

1

2

p(x)

−8

−1

0

1

8

The curve intersects xaxis at only one points (0,0), therefore, zeroes of p(x) =x^{3} is 0.
Let p(x) =x^{3} −x^{2} be any cubic polynomial.
x  −1  0  1  2 
p(x)  −2  0  0  4 
The curve intersects xaxis at two points (0,0) and (1,0), therefore, zeroes of p(x) =x^{3} −x^{2} are 0 and 1.
Relationship between Zeroes & Coefficients of a Polynomial
In a linear polynomial
p(x) = ax + b
If, ax + b = 0
⇒ x = −b/a
∴ Zero of a linear polynomial, ax + b = −b/a
= −Constant term/Coefficient of x .
In a quadratic polynomial
ax^{2} + bx + c = k(x−α )(x−β)
⇒ ax^{2} + bx + c = k{x^{2}−(α+β)x + α β}
⇒ ax^{2} + bx + c = kx^{2}−k(α+β)x + kαβ
Comparing the coefficients ofx^{2}, x and constant terms on both the sides, we get
a = k,
b = – k(α + β) and
c = kαβ
α +β= −b/a
αβ = c/a
Therefore,
Sum of zeroes
α + β = −b/a
= −(Coefficient of x )/Coefficient of x^{2},
Product of zeroes
αβ = c/a
= Constant term/Coefficient of x^{2} .