Relationship between Zeroes and Coefficients of a Polynomial (original) (raw)

Last Updated : 16 Apr, 2026

Polynomials are algebraic expressions with variables and coefficients. The zeroes of a polynomial are the values that make it equal to zero, and they have a specific relationship with its coefficients.

• **Zeros of a Polynomial****:** Values of the variable for which the polynomial becomes equal to zero.
• **Coefficients of a Polynomial: Numerical values that are multiplied with the variables in a polynomial expression.

The relationship between the zeroes and coefficients of a polynomial shows how the roots of a polynomial are connected to its coefficients.

• According to the Factor Theorem, if k is a zero of P(x), then (x − k) is a factor of the polynomial.
• Using this idea, formulas are derived that relate the sum and product of zeroes to the coefficients of the polynomial.

**Linear Polynomial

A linear polynomial is a polynomial of degree 1 and is written in the form ax + b, where x is a variable and a and b are constants (a ≠ 0). If P(x) = ax + b, then its zero is -b/a.

• Zero of a linear polynomial = − (constant term) / (coefficient of x)

**Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2 and is written in the form ax² + bx + c, where x is a variable and a, b, c are constants with a ≠ 0. Let α and β be the zeroes of the polynomial.

• Sum of zeroes (α + β) = −b/a = − (coefficient of x) / (coefficient of x²)
• Product of zeroes (αβ) = c/a = (constant term) / (coefficient of x²)

**Example: Find the zeros of the polynomial, P(x) = 2x2 - 8x + 6

**Solution:

P(x) = 2x2 -8x + 6

⇒ P(x) = 2x2 - 6x - 2x + 6

⇒ P(x) = 2x(x - 3) -2(x - 3)

⇒ P(x) = 2(x - 1)(x - 3)

So the zeroes of the polynomial are,

x - 1 = 0

**⇒ x = 1

And x - 3 = 0

**⇒ x = 3

• Sum of Zeros = 1 + 3 = 4
• Product of Zeros = 1 × 3 = 3

Using the relationship as discussed above.

Given equation,

2x2 -8x + 6 = 0 comparing with ax2 + bx + c = 0

a = 2, b = -8, and c = 6

• Sum of Roots = -b/a = -(-8)/2 = 8/2 = 4
• Production of the roots = c/a = 6/2 = 3

Thus, the relationship between the zeros of the quadratic polynomial and the coefficient of the quadratic polynomial holds true.

**Cubic Polynomial

A cubic polynomial is a polynomial of degree 3 and is written in the form ax³ + bx² + cx + d, where x is a variable and a, b, c, d are constants with a ≠ 0. Let α, β, γ be the zeroes of the polynomial.

• Sum of zeroes (α + β + γ) = −b/a = − (coefficient of x²) / (coefficient of x³)
• Sum of product of zeroes (αβ + βγ + αγ) = c/a = (coefficient of x) / (coefficient of x³)
• Product of zeroes (αβγ) = −d/a = − (constant term) / (coefficient of x³)

Solved Examples

**Example 1: Find the sum of the roots and the product of the roots of the polynomial x3 -2x2 - x + 2.

**Solution:

Given Polynomial,

x3 -2x2 - x + 2

comparing with ax3 + bx2 + cx + d = 0

a = 1, b= -2, c = -1, and d = 2

Sum of the roots (p + q+ r) = – Coefficient of x2/ coefficient of x3

= -b/a
= -(-2)/1 = 2

Product of the roots (pqr) = – Constant Term/Coefficient of x3

= -d/a
= -2/1 = -2

**Example 2: Find the sum and product of the zeros of the quadratic polynomial 6x2 + 18 = 0

**Solution:

Given Polynomial 6x2 + 18 = 0

It can be also written as, 6x2 + 0x + 18 = 0

Comparing with ax2 + bx + c = 0

a = 6, b = 0, and c = 18

Sum of Zeroes = – Coefficient of x/ Coefficient of x2

= -b/a

= -0/6
= 0

Product of the Zeroes = Constant term / Coefficient of x2

= c/a
= 18/6
= 3

**Example 3: For the given polynomial ax2 + bx + 1 = 0. Its roots are -1 and 3. Find the values of a and b.

**Solution:

Let m and n be the roots of the quadratic equation ax2 + bx + 1 = 0

Here,

• m = -1
• n = 3

We know that,

m + n = -b/a

**⇒ -1 + 3 = -b/a

**⇒ -b/a = 2...(i)

And m.n = c/a

**⇒ (-1)(3) = 1/a

**⇒ -3 = 1/a

**⇒ a = -1/3...(ii)

from (i) we get,

-b/a = 2

**⇒ b = -2a

**⇒ b = -2(-1/3) = 2/3