Relationship between Zeroes and Coefficients of a Polynomial

Relationship between Zeroes and Coefficients of a Polynomial

 You have already seen that zero of a linear polynomial ax + b is

 We will now try to answer the question raised in Section 2.1 regarding the relationship between zeroes and coefficients of a quadratic polynomial. For this, let us take a quadratic polynomial, say p(x) = 2x² – 8x + 6. In Class IX, you have learnt how to factorise quadratic polynomials by splitting the middle term. So, here we need to split the middle term ‘– 8x’ as a sum of two terms, whose product is 6 × 2x² = 12x². So, we write

2x² – 8x + 6 = 2x² – 6x – 2x + 6 = 2x(x – 3) – 2(x – 3)

                    = (2x – 2)(x – 3) = 2(x – 1)(x – 3)

So, the value of p(x) = 2x² – 8x + 6 is zero when x – 1 = 0 or x – 3 = 0, i.e., when x = 1 or x = 3. So, the zeroes of 2x² – 8x + 6 are 1 and 3. Observe that:
 

 
Let us take one more quadratic polynomial, say, p(x) = 3x² + 5x – 2. By the method of splitting the middle term,

    
 Hence, the value of 3x² + 5x – 2 is zero when either 3x – 1 = 0 or x + 2 = 0, i.e.,

when x=1/3 or x =-2.   So, the zeroes of  Observe that:

 

In general, if α* and β* are the zeroes of the quadratic polynomial p(x) = ax² + bx + c,

a≠0,  then you know that x-α and x-β     are the factors of p(x). Therefore,

 where k is a constant

 

                             

Comparing the coefficients of x², x and constant terms on both the sides, we get
   

 
                          

 

Division algorithm for Polynomials

 You know that a cubic polynomial has at most three zeroes. However, if you are given only one zero, can you find the other two? For this, let us consider the cubic polynomial

x3-3x2-x+3.  If we tell you that one of its zeroes is 1, then you know that x – 1 is a factor of x³ – 3x² – x + 3. So, you can divide x³ – 3x² – x + 3 by x – 1, as you have learnt in Class IX, to get quotient x² – 2x – 3.

Next, you could get the factors of x² – 2x – 3, by splitting the middle term, as (x + 1) (x – 3). This word give you

 

    

So, all the three zeroes of the cubic polynomial are now known to you as 1, – 1, 3.
Let us discuss the method of dividing one polynomial by another in some detail.
​​​​​​​Before noting the steps formally.

 then we can find polynomials q(x) and r(x) such that
 
Where r(x) = 0 or degree of r (x) < degree of g(x).