Ex

  Mathematics (NTSE/Olympiad)  

Polynomial

Geometric Meaning of the Zeros of a Polynomial

To Understand the concept of Geometric Meaning of the Zeros of a Polynomial. Let us consider linear polynomial ax + b. The graph of y = ax + b is a straight line.
For example : The graph of y = 3x + 4 is a straight line passing through (0, 4) and (2, 10).




(i) Let us consider the graph of y = 2x – 4 intersects the x-axis at x = 2. The zero 2x – 4 is 2. Thus, the zero of the polynomial 2x – 4 is the x-coordinate of the point where the graph y = 2x – 4 intersects the x-axis.


(ii) A general equation of a linear polynomial is ax + b. The graph of y = ax + b is a straight line which intersects the x-axis at Zero of the polynomial ax + b is the x-coordinate of the point of intersection of the graph with x-axis.

(iii) Let us consider the quadratic polynomial x2 – 4x + 3. The graph of x2 – 4x + 3 intersects the x-axis at the point (1, 0) and (3, 0).
Zeroes of the polynomial x2 – 4x + 3 are the x-coordinates of the points of intersection of the graph with x-axis.

The shape of the graph of the quadratic polynomials is ∪ and the curve is known as parabola.

(iv) Now let us consider one more polynomial –x2 + 2x + 8. Graph of this polynomial intersects the x-axis at the points (4, 0), (–2, 0). Zeroes of the polynomial –x2 + 2x + 8 are the x-coordinates of the points at which the graph intersects the x-axis. The shape of the graph of the given quadratic polynomial is ∩ and the curve is known as parabola.


The zeroes of a quadratic polynomial ax2 + bx + c he x-coordinates of the points where the graph of y = ax2 + bx + c intersects the x-axis.
Cubic polynomial : Let us find out geometrically how many zeroes a cubic has. To understand this let consider cubic polynomial x3 – 6x2 + 11x – 6.

Case 1 : The graph of the cubic equation intersects the x-axis at three points (1, 0), (2, 0) and (3, 0). Zeroes of the given polynomial are the x-coordinates of the points of intersection with the x-axis.

Case 2 : The cubic equation x3 – x2 intersects the x-axis at the point (0, 0) and (1, 0). Zero of a polynomial x3 – x2 are the x-coordinates of the point where the graph cuts the x-axis. Zeroes of the cubic polynomial are 0 and 1.

Case 3 : Let us consider the equation of cubic polynomial y = x3. Cubic polynomial has only one zero.

In short, we say that, a cubic equation can have 1 or 2 or 3 zeroes or any polynomial of degree three can have at most three zeroes.

Important Note: In general, polynomial of degree n, the graph of y = p(x) passes x-axis at most at n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.


If you want to give information about online courses to other students, then share it with more and more on Facebook, Twitter, Google Plus. The more the shares will be, the more students will benefit. The share buttons are given below for your convenience.
×

NTSE Mathematics (Class X)

  • Trigonometry
  • Similar Triangles
  • Statistics
  • Quadratic Equation
  • Arithmetic Progressions
  • Application of Trigonometry
  • Circle
  • Co-ordinate Geometry
  • Area related to Circle
  • Surface Area & Volume
  • Constructions
  • Probability

NTSE Mathematics (Class IX)

  • Trigonometry
  • Similar Triangles
  • Statistics
  • Quadratic Equation
  • Arithmetic Progressions
  • Application of Trigonometry
  • Circle
  • Co-ordinate Geometry
  • Area related to Circle
  • Surface Area & Volume
  • Constructions
  • Probability

SHOW CHAPTERS

NTSE Physics Course (Class 9 & 10)

NTSE Chemistry Course (Class 9 & 10)

NTSE Geography Course (Class 9 & 10)

NTSE Biology Course (Class 9 & 10)

NTSE Democratic Politics Course (Class 9 & 10)

NTSE Economics Course (Class 9 & 10)

NTSE History Course (Class 9 & 10)

NTSE Mathematics Course (Class 9 & 10)