Relationship Between Zeroes and Coefficients of Polynomials
The number of zeroes of a polynomial equals the degree of the polynomial, and there is a well-defined mathematical relationship between the zeroes and the coefficients. In this lesson, let's explore the relationship between the zeroes and coefficients of a polynomial.
Definition of Zeroes of a Polynomial
Zeroes of a polynomial are the solutions to the given polynomial equation when the polynomial is set as equal to zero. Polynomials are classified depending on the highest power of the variable in the given polynomial. Mathematically, if p(x) is a polynomial with variable x, and k is any real number and said to be the zero of the polynomial p(x) if p(x) at x = k is 0.
Definition of Coefficients of a Polynomial
A coefficient is a number or quantity placed with a variable, usually an integer that is multiplied by the variable next to it. For the variables with no integer with them are assumed to have 1 as their coefficient. A coefficient can be positive or negative, real or imaginary, or in the form of decimals or fractions.
Relationship Between Zeroes and Coefficients of a Polynomial
The relationship between zeroes and the coefficients of polynomials can be defined based on the definite formulas as per the type of polynomial. The relation between the zeroes and the coefficients of a polynomial is given below:
Linear Polynomial
A linear polynomial is an expression of the form ax + b, having 1 as the degree of the polynomial. Here, “x” is a variable, “a” and “b” are constants. The zero of the polynomial = -b/a = – constant term/coefficient of x.
Quadratic Polynomial
The Quadratic polynomial is an expression of the form ax2 + bx + c having the highest degree 2. Here, “x” is a variable, “a”, "b", and “c” are constants and a ≠ 0. Let α and β be the two zeroes of the polynomial, then
- The sum of zeroes, α + β is -b/a = – Coefficient of x/ Coefficient of x2
- The product of zeroes, αβ is c/a = Constant term / Coefficient of x2
Cubic Polynomial
The cubic polynomial is an expression of the form ax3 + bx2 + cx + d having the highest degree 3. Here, “x” is a variable, “a”, "b", and “c” are constants, and a ≠ 0. Let α, β, and γ are the three zeroes of the polynomial, then
- The sum of zeroes, α + β + γ is -b/a = – Coefficient of x2/ coefficient of x3
- The sum of the product of zeroes, αβ+ βγ + αγ is c/a = Coefficient of x/Coefficient of x3
- The product of zeroes, αβγ is -d/a = – Constant term/Coefficient of x3
Related Topics
Given below is a list of topics related to the relationship between zeroes and coefficients of polynomial.
Practice Questions on Relationship Between Zeroes and Coefficients of Polynomials
FAQs on Relationship Between Zeroes and Coefficients of Polynomials
What Is the Relationship Between Zeroes and Coefficients of Polynomials in Algebra?
The number of zeroes of a polynomial is determined by the degree of the polynomial, and thus there is a well-defined mathematical relationship between the zeroes and the coefficients.
What Is the Relationship Between Zeroes and Coefficients of a Cubic Polynomial?
The relationship between zeroes and coefficients of a cubic polynomial is as follows:
- The sum of zeroes = -b/a = – Coefficient of x2/ coefficient of x3
- The sum of the product of zeroes = c/a = Coefficient of x/Coefficient of x3
- The product of zeroes -d/a = – Constant term/Coefficient of x3
What Is the Relationship Between Zeroes and Coefficients of a Quadratic Polynomial?
The relationship between zeroes and coefficients of a quadratic polynomial is as follows:
- The sum of zeroes = -b/a = – Coefficient of x/ Coefficient of x2
- The product of zeroes = c/a = Constant term / Coefficient of x2
What Is the Relationship Between Zeroes and Coefficients of a Linear Polynomial?
The relationship between zeroes and coefficients of a linear polynomial is given as the zero of a polynomial = -b/a = – constant term/coefficient of x.