Ex

  Mathematics (NTSE/Olympiad)  

Polynomials

Relation between the zero and the coefficients of a polynomials

Consider quadratic polynomial P(x) = 2x2 – 16x + 30.
Now, 2x2 – 16x + 30 
= (2x – 6) (x – 3) 
= 2(x – 3) (x – 5)
The zeroes of P(x) are 3 and 5.

Sum of the zeroes are = 3 + 5 = 8 =

Product of the zeroes are = 3 × 5 = 15 = 

So if ax2 + bx + c, a ≠ 0 is a quadratic polynomial and a, b are two zeroes of polynomial then 

Example: Find the zeroes of the quadratic polynomial 6x2 – 13x + 6 and verify the relation between the zeroes and its coefficients.

Solution: We have,
    6x2 – 13x + 6
= 6x2 – 4x – 9x + 6
= 2x (3x – 2) –3 (3x – 2)
= (3x – 2) (2x – 3)
So, the value of quadratic equation 6x2 – 13x + 6 is 0, when
(3x – 2) = 0 or (2x – 3) = 0 i.e.,
When x =  or 
Therefore, the zeroes of 6x2 – 13x + 6 are  and .

Sum of the zeroes : 

 +  =  = 

Product of the zeroes
= ×  = =

Example: Find the zeroes of the quadratic polynomial 4x2 – 9 and verify the relation between the zeroes and its coefficients.

Solution: We have,
     4x2 – 9
= (2x)2 – 32
= (2x – 3) (2x + 3)
So, the value ofa quadratic equation 4x2 – 9 is 0, when
2x – 3 = 0 or 2x + 3 = 0
i.e., when x = or x = .
Therefore, the zeroes of 4x2 – 9 are  & .

Sum of the zeroes 
= + = 0 = =

Product of the zeroes
= =

Example: Find the zeroes of the quadratic polynomial 9x2 – 5 and verify the relation between the zeroes and its coefficients.

Solution: We have,
 9x2 – 5 
= (3x)2 – (√5)2
= (3x – √5) (3x +√5)
So, the value of 9x2 – 5 is 0, when 
3x –√5 = 0 or 3x +√5 = 0
i.e., when x =  or x = .

Sum of the zeroes

== 0 = =

Product of the zeroes
= ×=


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NTSE Mathematics (Class X)

  • Trigonometry
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  • Quadratic Equation
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NTSE Mathematics (Class IX)

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  • Similar Triangles
  • Statistics
  • Quadratic Equation
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  • Co-ordinate Geometry
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  • Surface Area & Volume
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