Relationship between Zeros and Coefficients of a Polynomial

The relationship between zeros and coefficients of a polynomial is a fundamental concept in algebra that links the roots of a polynomial to its coefficients. It helps understand how a polynomial is formed and makes it easier to verify solutions or build equations without actually finding the zeros every time. This concept is widely used in solving quadratic and higher-degree polynomial problems in mathematics. In this guide, you will learn about the relationship between the zeros and coefficients of a polynomial and its formulas with easy steps and examples.

Table of Contents:

• What are Coefficients of a Polynomial
• What are Zeros of a Polynomial
• The Relationship Between Zeros and Coefficients of a Polynomial
• Solved Examples on The Relationship Between Zeros and Coefficients of a Polynomial

What are  Coefficients of a Polynomial

A polynomial in one variable, x, is an algebraic expression of the form:
p(x) = anxn+an−1xn−1+...+a1x+a0

where an,an−1,...,a0 are real numbers,  an≠0, and n is a non-negative integer (called the degree).

A coefficient is the numerical value (or constant) that multiplies the variable in each term of the polynomial.

Coefficients can be positive, negative, integers, or fractions.

For example, in the polynomial p(x) = 4x2−7x+3:

• The coefficient of x² is 4

• The coefficient of x is −7

• The constant term is 3 (this is the coefficient of x⁰)
  
  

Read more: Quadratic Equations

What are Zeros of a Polynomial

A "zero" of a polynomial is the value of x that makes the entire polynomial equal to zero.
A real number 'k' is a zero of the polynomial p(x) if p(k) = 0

For example, in the polynomial p(x) =  x2−5x+6

p(2)=(2)2−5(2)+6=4−10+6=0.

∴ 2 is a zero of the polynomial  x2−5x+6

 p(3)=(3)2−5(3)+6=9−15+6=0.
∴ 3 is also a zero of the polynomial   x2−5x+6

The Relationship Between Zeros and Coefficients of a Polynomial

The relationship between zeroes and coefficients of a polynomial is a set of formulas called Vieta's Formulas that connect the sum and product of the zeroes directly to the coefficients of the polynomial, without you ever needing to find the zeroes.

1. Linear Polynomial:

Standard form: p(x) = ax + b, where a ≠ 0

A linear polynomial has exactly one zero.

ax + b = 0
ax = −b
x = −b/a
So the zero of a linear polynomial is x =  −ba=−Constant termCoefficient of x

2. Quadratic Polynomial

Standard form :p(x)=ax2+bx+c,where a ≠ 0

A quadratic polynomial has exactly two zeroes (which may be real and distinct, real and equal, or complex). Let us call them α (alpha) and β (beta).

• Sum of Zeros

 Let α and β be zeros of the polynomial  ax2+bx+c;
  α + β = −b/a

 The sum of the zeroes equals the negative of the coefficient of x, divided by the coefficient of x².

• Product of Zeros
  Let α and β be zeros of the polynomial  ax2+bx+c;
  αβ = c/a

The product of the zeroes equals the constant term divided by the coefficient of x².

3. Cubic Polynomial

Standard form: p(x) =  ax3+bx2+cx+d, where a ≠ 0

A cubic polynomial has exactly three zeroes (which may be real and distinct, real and equal, or complex). Let us call them α (alpha), β (beta), and γ (gamma).

• Sum of Zeros

Let α, β, and γ be zeros of the cubic polynomial ax3+bx2+cx+d

α + β + γ = −b/a

That is, the sum of zeroes = − (Coefficient of x²) / (Coefficient of x³)

• Product of Zeros taken two at a time
  Let α, β, and γ be zeros of the cubic polynomial  ax3+bx2+cx+d
  αβ + βγ + γα = c/a
  That is, sum of pairwise products = (Coefficient of x) / (Coefficient of x³)

• Product of all three zeros
  αβγ = −d/a

That is, product of all zeroes = − (Constant term) / (Coefficient of x³)

Solved Examples on The Relationship Between Zeros and Coefficients of a Polynomial

Example 1: Find the zero of p(x) = 5x − 10.

Solution: Given a = 5 and b = −10.

Zero = −b/a = −(−10)/5 = 10/5 = 2

Verification: p(2) = 5(2) − 10 = 10 − 10 = 0 

Example 2: Find the sum and product of the zeroes of the polynomial p(x) = 2x² + 5x + 3.

Solution: Given a = 2, b = 5, c = 3.

Sum of zeroes = (α + β) = −b/a = −5/2

Product of zeroes  = (αβ) = c/a = 3/2

Verification: 2x² + 5x + 3 = 2x² + 2x + 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3) (x + 1)

Zeros are x = −3/2 and x = −1.
Sum = −3/2 + (−1) = −3/2 − 2/2 = −5/2
Product = (−3/2) × (−1) = 3/2 

Example 3: If the sum of zeroes is 4 and the product of zeroes is 3, form the quadratic polynomial.
Solution: Given Sum of zeroes = (α + β) = −b/a = 4
Product of zeroes  = (αβ) = c/a = 3
A quadratic polynomial with zeroes α and β can always be written as x² − (α + β)x + αβ = x² − 4x + 3

Example 4: Verify the relationship between zeroes and coefficients for p(x) = x³ − 4x² + 5x − 2, given that the zeroes are 1, 1, and 2.
Solution: Given p(x) = x³ − 4x² + 5x − 2,  and zeroes are 1, 1, and 2.
Let α = 1, β = 1, γ = 2. Here, a = 1, b = −4, c = 5, and d = −2.
Sum of zeroes:  α + β + γ = 1 + 1 + 2 = 4.
−b/a = −(−4)/1 = 4
Sum of products taken two at a time: αβ + βγ + γα = (1)(1) + (1)(2) + (2)(1) = 1 + 2 + 2 = 5
c/a = 5/1 = 5
Product of zeroes: αβγ = (1)(1)(2) = 2
−d/a = −(−2)/1 = 2 

Hence, all the relations are verified.