In Mathematics, algebraic expression with many terms is called a polynomial. We know that, depending on the degree of the polynomial, the polynomial can be classified into different types, such as linear polynomial (x), quadratic polynomial (x²), and cubic polynomial (x³), and so on. From the degree of the polynomial, we can easily determine the zeros of a polynomial. Because the number of zeros of the polynomial expression is equal to the polynomial degree. In this article, let us discuss the relationship between the zeros and coefficients of a polynomial with more solved examples.

Relationship Between the Zeros and Coefficients of a Polynomial

A real number say “a” is a zero of a polynomial P(x) if P(a) = 0. The zero of a polynomial is clearly explained using the Factor theorem. If “k” is a zero of a polynomial P(x), then (x-k) is a factor of a given polynomial. The relation between the zeros and the coefficients of a polynomial is given below:

Linear Polynomial

The linear polynomial is an expression, in which the degree of the polynomial is 1. The linear polynomial should be in the form of ax+b. Here, “x” is a variable, “a” and “b” are constant. 

The polynomial P(x) is ax+b, then the zero of a polynomial is -b/a = – constant term/coefficient of x)

Quadratic Polynomial

The Quadratic polynomial is defined as a polynomial with the highest degree of 2. The quadratic polynomial should be in the form of ax² + bx + c. In this case, a ≠ 0. Let say α and β are the two zeros of a polynomial, then 

The sum of zeros, α + β is -b/a = – Coefficient of x/ Coefficient of x²

The product of zeros, αβ is c/a = Constant term / Coefficient of x²

Cubic Polynomial

The cubic polynomial is a polynomial with the highest degree of 3. The cubic polynomial should be in the form of ax³ + bx² + cx + d, where a ≠ 0. Let say α, β, and γ are the three zeros of a polynomial, then

The sum of zeros, α + β + γ is -b/a = – Coefficient of x²/ coefficient of x³

The sum of the product of zeros, αβ+ βγ + αγ is c/a = Coefficient of x/Coefficient of x³

The product of zeros, αβγ is -d/a = – Constant term/Coefficient of x³

Zeros of a Polynomial Solved Examples

Example: 

Evaluate the sum and product of zeros of the quadratic polynomial 4x² – 9.

Solution:

Given quadratic polynomial is 4x² – 9.

 4x² – 9 can be written as 2x² – 3³, which is equal to (2x+3)(2x-3).

To find the zeros of a polynomial, equate the above expression to 0

(2x+3)(2x-3) = 0

2x+3 = 0

2x = -3

X = -3/2

Similarly, 2x-3 = 0,

2x = 3

x = 3/2

Therefore, the zeros of a given quadratic polynomial is 3/2 and -3/2.

Finding the sum and product of a polynomial:

The sum of the zeros = (3/2)+ (-3/2) = (3/2)-(3/2) = 0 

The product of zeros = (3/2).(-3/2) = -9/4.