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.
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)