RELATIONSHIP BETWEEN ZEROS AND COEFFICIENT OF A POLYNOMIAL

RELATIONSHIP  BETWEEN ZEROS AND COEFFICIENT OF A POLYNOMIAL

relationship  between zeros and coefficient of a polynomial in case of quadratic and cubic polynomial is stated as follows

(1) QUADRATIC POLNOMIAL

Let ax² +bx +c be the quadratic polynomial and α and β are its zeros ,then

Sum of zeros = α + β = -b/a = - (coefficient of x)/ (coefficient of x²)

Product of zeros = αβ = c/a = constant term / (coefficient of x²)

If we need to form an equation of degree two ,when sum and products of the roots is given ,then K[x²-( α + β )x + αβ ]=0 is the required equation ,where k is constant .

Procedure for finding zeros of a quadratic polynomial  

·         Find  the factors of the quadratic polynomial .

·         Equate each of the above factors (step 1) with zero.

·         Solve the above equation (step 2)  

·         The value of the variables obtained (step 3) are the required zeros .

(2) CUBIC POLYNOMIAL

Let axᶟ +bx² +cx +d  be the cubic polynomial and α , β and γ are its zeros ,then

 Sum of zeros = α + β + γ = -b/a = - (coefficient of x²)/ (coefficient of xᶟ)

Sum of Product of zeros taken two at a time  = αβ +βγ +γα = c/a =  (coefficient of x)/ (coefficient of xᶟ)

Product of zeros = αβγ = -d/a = - constant term / (coefficient of xᶟ)

When sum of zeros , Sum of Product of zeros taken two at a time , Product of zeros is given , then K[xᶟ-( α + β + γ )x² + (αβ +βγ +γα)x – αβγ ]=0 is the required equation ,where k is constant,

Procedure for finding zeros of a cubic polynomial

·         By  hit and trial method find one zeros of the polynomial using remainder theorem

·         Now if we know one zero , then we know one factor of the polynomial . divide the cubic polynomial by this factor to obtain quadratic polynomial

·         Now , solve this quadratic polynomial to obtain the other two zeros of the cubic polynomial .

·         These three zeros are the required zreos