Class 10 Maths Polynomials Relationship Zeroes & Coefficients of Polynomia

Relationship between Zeroes and Coefficients of a Polynomial

We know that zero of a linear polynomial ax + b is –b/a

Let’s find the relationship between zeroes and coefficients of a quadratic polynomial.

For a quadratic polynomial p(x) = ax2 + bx + c, if the roots are α & β, then α + β = -b/a  & α * β = c/a

For Proof, refer ExamFear video lessons on this chapter.

 

Numerical: Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Solution:  p(x)=  x2 + 7x + 10= (x + 2)(x + 5)

p(x) = 0, when x= -2 or x= -5, therefore zeroes of p(x)=  x2 + 7x + 10 are -2 & -5.

Let’s see if α + β = -b/a  & α * β = c/a here a=1, b=7 & c =10

 

Sum of Zeroes = α + β = -2 + -5  = -7         -(i)

-b/a = -7/1  =  -7                                                -(ii)

Comparing i & ii,               α + β = -b/a 

 

Product of Zeroes = α * β   =  -2 *  -5   = 10            - (iii)

c/a = 10/1  = 10                                                                  - (iv)

Comparing iii & iv,            α * β = c/a

 

Numerical: Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.

Solution:   General form of quadratic polynomial is  p(x) = ax2 + bx + c   -(i)

α + β = -b/a  =  -3    or b = 3a                        -(ii)

& α * β = c/a =  2    or c = 2a                          -(iii)

 

Let a=k, then b=3k & c= 2k,    using (ii) & (iii)

Thus the equation will be  p(x) = kx2 + 3kx + 2k   or  p(x)  = k(x2 + 3x + 2)   where k is any real number.

 

 

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