Notes On Relationship between Zeroes and Coefficients of a Polynomial - CBSE Class 10 Maths
A polynomial is an algebraic expression consisting of multiple terms. There are various types of polynomials such as linear, quadratic, cubic and so on.

A real number k is a zero of a polynomial of p(x) if p(k) = 0.

Factor Theorem:
If a is zero of a polynomial p(x) then (x – a) is a factor of p(x).

Relationship betweeen Zeroes and coefficients of a Polynomial

The general form of linear polynomial is p(x) = ax+b, its zero is  -b a .i.e.x =  -b a or - Constant term     Coefficient of x   .

General form of quadratic polynomial is ax2 + bx +c where a ≠ 0. There are two zeroes of quadratic polynomial.

Sum of zeroes =  -b a = - Coefficient of  x       Coefficient of   x 2  

Product of zeroes = c a   Constant term       Coefficient of   x 2   .

General form of cubic polynomial of ax3 + bx 2+ cx + d where a ≠ 0. There are three zeroes of cubic polynomial.

The sum of zeroes of the cubic polynomial =  -b a - coefficient of  x 2 Coefficient of  x 3  

Sum of the product of zeroes taken two at a time =  c a coefficient of  x  Coefficient of  x 3   
 
Product of zeroes =  -d a - Constant term   Coefficient of  x 3   .

Summary

A polynomial is an algebraic expression consisting of multiple terms. There are various types of polynomials such as linear, quadratic, cubic and so on.

A real number k is a zero of a polynomial of p(x) if p(k) = 0.

Factor Theorem:
If a is zero of a polynomial p(x) then (x – a) is a factor of p(x).

Relationship betweeen Zeroes and coefficients of a Polynomial

The general form of linear polynomial is p(x) = ax+b, its zero is  -b a .i.e.x =  -b a or - Constant term     Coefficient of x   .

General form of quadratic polynomial is ax2 + bx +c where a ≠ 0. There are two zeroes of quadratic polynomial.

Sum of zeroes =  -b a = - Coefficient of  x       Coefficient of   x 2  

Product of zeroes = c a   Constant term       Coefficient of   x 2   .

General form of cubic polynomial of ax3 + bx 2+ cx + d where a ≠ 0. There are three zeroes of cubic polynomial.

The sum of zeroes of the cubic polynomial =  -b a - coefficient of  x 2 Coefficient of  x 3  

Sum of the product of zeroes taken two at a time =  c a coefficient of  x  Coefficient of  x 3   
 
Product of zeroes =  -d a - Constant term   Coefficient of  x 3   .

Videos

References

Next