Relationship Between Zeroes And Coefficients Of A Polynomial

 

Any algebraic expression like 4x+2, 2y2 -3y +4, 5x3 -4x2 + x -2 etc are called polynomials.

4x+2 is a polynomial in the variable x of degree1 and is called linear polynomial.

2y2-3+4 is a polynomial in the variable y of degree 2 and is called quadratic polynomial.

5x3-4x2 +x -2 is a polynomial in the variable x of degree 3 and is called cubic polynomial and so on.

Now in general let us take a quadratic polynomial as ax2 + bx +c. Here coefficient of x2 and x is 'a' and 'b' respectively. 'c' is a constant term.

Now let us understand what is 'zero of polynomial' means?

If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).

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

 

In linear the relationship between zeros and coefficient of a polynomial

The general form of linear polynomial is p(x)  = ax +b , its zero is -b/a or minus of constant term divided by coefficient of x.

If 'α' is the zero of the above linear polynomial then,

α = -b/a

 

n quadratic the relationship between zeros and coefficient of polynomial

In the general form of quadratic polynomial ax2 + bx + c, there are two zeros say α and β, then;

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

Product of zeros = α.β = c/a = (Constant Term) / (Coefficient of x2)

 

Example on relationship between zeros and coefficient of a polynomial

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

We have,  

 x2 + 7x +10 = (x +2) (x+5)

So, the value of x2 + 7x + 10 is zero,

when x+2 = 0 or x+5 = 0, i.e.,

when, x = -2 or x = -5.

Therefore, the zeroes of x2 + 7x + 10 are -2 and -5. 

Now, the relationship between zeros and coefficient of above polynomial can be shown as:-

Sum of zeroes = -2 + (-5) = -7 = -(7)/1 = -(coefficient of x) / (coefficient of x2)

Product of zeroes = (-2) x (-5) = 10 =10/1 = (constant Term) / (coefficient of x2)