Graph of Polynomials

 

The value of a polynomial can be different for different values of variables in it.

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 x-axis and the polynomial is represented along y-axis. 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 x-axis 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 x-axis at exactly one point, (−b/a  , 0).

∴ The linear polynomial ax + b, a ≠ 0, has exactly one zero, the x-coordinate of the point where the graph of p(x) = ax + b intersects the x-axis.

Graph of a Quadratic polynomial

General form of a quadratic polynomial

ax2 + bx + c

Let p(x) = 2x2 + 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 x-axis 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 ax2 + bx + c, a ≠ 0, the graph is always a parabola but it’s position and orientation can be different.

Case I ↬

The graph cuts x-axis at two distinct points Q’ and Q.
The x-coordinates of Q’ and Q are the two zeroes of the quadratic polynomial, ax2 + bx + c in this case.
Case II ↬

The graph cuts x-axis at exactly one point, i.e., two points Q’ and Q of Case (i) coincide here to become one point Q.
The x-coordinate of Q is the only zero for the quadratic polynomial, ax2 + bx + c in this case.
Case III ↬

The graph is either completely above the x-axis or completely below the x-axis. So, it does not cut the x-axis at any point.
So, the quadratic polynomial, ax+ 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

ax3 + bx2 + cx + d

Let p(x) = x3 − 4x  be any cubic polynomial.

x
−2
−1
0
  1
2
p(x)
  0
  3
0
−3
0

The curve intersects x-axis at three points  (-2,0), (0,0) and (2,0) therefore, zeroes of p(x) =x3 − 4x are −2, 0 and 2.

Let p(x) =x3  be any cubic polynomial.

x
−2
−1
0
1
2
p(x)
−8
−1
0
1
8

The curve intersects x-axis at only one points  (0,0), therefore, zeroes of p(x) =x3  is 0.

Let p(x) =x3  −x2  be any cubic polynomial.

x −1 0 1 2
p(x) −2 0 0 4

The curve intersects x-axis at two points (0,0) and (1,0), therefore, zeroes of p(x) =x3 −x2 are 0 and 1.

∴ A polynomial of degree 3 can have at most three zeroes.

Relationship between Zeroes & Coefficients of a Polynomial

In a linear polynomial

p(x) = ax + b

If, a+ b = 0

x = −b/a

∴ Zero of a linear polynomial, ax + b = −b/a

= −Constant term/Coefficient of x .

In a quadratic polynomial

p(x) = ax2 + bx + c = 0
If, ax2 + bx + c = 0
⇒ k(x−α )(x−β) = 0
(where (x−α ) & (x−β) are factors of p(x) & k is a constant).
x = α , β
⇒ α  and β are zero of a quadratic polynomialax2 + bx + c .
So,

ax2 + bx + c = k(x−α )(x−β)

⇒ ax2 + bx + c  = k{x2−(α+β)x + α β}
⇒ ax2 + bx + c  = kx2−k(α+β)x + kαβ
Comparing the coefficients ofx2, 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 x2,

Product of zeroes

αβ = c/a

= Constant term/Coefficient of x2 .

 

In a Cubic polynomial

p(x) = ax3 + bx2 + cx + d
If α, β, γ are the zeroes of the polynomial , then
α + β + γ = –b/a
= −(Coefficient of x2)/Coefficent of x3,
αβ + βγ + γα = c/a
= (Coefficient of x)/Coefficent of x3,
α β γ = −d/a
= −(Constant term)/Coefficent of x3.
A cubic polynomial has at most three zeroes.
If we are given only one zero,  we can find the other two.
First we divide the polynomial by the known zero, then the quotient so obtained is factorised by splitting it’s middle term to get the other two zeroes.

Division Algorithm for Polynomials

The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that,
p(x) = g(x) q(x) + r(x),
where, r(x) = 0 or degree r(x) < degree g(x).

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