Any algebraic expression like 4x+2, 2y^{2} -3y +4, 5x^{3} -4x^{2} + x -2 etc are called polynomials.

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

2y^{2}-3+4 is a polynomial in the variable y of degree 2 and is called **quadratic polynomial.**

5x^{3}-4x^{2} +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 ax^{2} + bx +c. Here coefficient of x^{2} 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 ax^{2} + bx + c, there are two zeros say α and β, then;

Sum of the zeros = **α + β = -b/a = -(coefficient of x) / (coefficient of x ^{2})**, and

Product of zeros = **α.β = c/a = (Constant Term) / (Coefficient of x ^{2})**

Example on relationship between zeros and coefficient of a polynomial

**Example:** Find the zeros of the quadratic polynomial x^{2} + 7x +10, and verify the relationship between the zeroes and the coefficients.

We have,

x^{2} + 7x +10 = (x +2) (x+5)

So, the value of x^{2} + 7x + 10 is zero,

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

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

Therefore, the zeroes of x^{2} + 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 x^{2})

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