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