Relationship between Zeroes and Coefficients of a Polynomial

Polynomials are algebraic expressions with constants and variables that can be linear i.e. the highest power o the variable is one, quadratic and others. The zeros of the polynomials are the values of the variable (say x) that on substituting in the polynomial give the answer as zero.

While the coefficients of a polynomial are the constants that are multiplied by the variables of the polynomial. There is a relation between the Zeroes of a Polynomial and the Coefficients of a Polynomial which is widely used in solving problems in algebra.

In this article, we will learn about the Zeroes of a Polynomial, the Coefficients of a Polynomial, and their relation in detail.

Zeroes of a Polynomial

For any polynomial f(x) zeros of the polynomial are defined as the values of the x which substituting in f(x) results in the zero value of the polynomial. They are also called the solution of the polynomial. Mathematically we define the zero of the polynomial as for any polynomial f(x) a is the zero of the polynomial if 

f(a) = 0

For example, if the given polynomial is f(x) = x+3 then -3 is the zero of the polynomial as,

f(-3) = -3+3 =0

The number of zeroes of the polynomial depends on the degree of the polynomial, i.e. for a linear polynomial (polynomial with degree 1) we have 1 zero, for a quadratic polynomial (polynomial with degree 2) we have 2 zeros, and so on.

Coefficients of a Polynomial

A polynomial is a function of x and its powers. It can be linear, quadratic, and others. A coefficient of a polynomial is the numerical value related to each x and its power in the polynomial.

For example, in the linear polynomial ax + b the coefficient of x is 'a' similarly, in px² + qx + r the coefficient of x²  is 'p' and the coefficient of x is 'q'

The coefficient of a polynomial can be any real number, fractions, or decimals and they can also be imaginary numbers.

Relationship between Zeros and Coefficients of a Polynomial

We know that zeros of any polynomial are the points where the graph of the polynomial cuts the x-axis. These zeros can also be found using the coefficients of different terms in a polynomial. Let's look at the relationship between the zeros and coefficients of the polynomials for various types of polynomials.

The following image shows the relationship between Zeros and Coefficients of a Polynomial.

Linear Polynomial

A linear polynomial, in general, is defined by,

y = ax + b

We know, for zeros we need to find the points at which y = 0. Solving this general equation for y = 0. 

y = ax + b 

⇒ 0 = ax + b  

⇒ x = -b/a

This gives us the relationship between zero and the coefficient of a linear polynomial. 

In general for a linear equation y = ax + b, a ≠ 0, the graph of ax + b is a straight line that cuts the x-axis at (-b/a, 0)

Example: Find the zeros of the linear polynomial.

y = 4x + 2

Solution:

Given Equation y = 4x + 2

Here, 

• a = 4
• b = 2

So, by the formula mentioned above the zero will occur at (-b/a, 0) that is (-2/4, 0) or (-1/2, 0)

Let's verify this zero using Graphical Method.

Given Equation

y = 4x + 2 

In intercept Form

x/(-1/2) + y/(2) = 1

Now we know the intercepts on the x and y-axis. 

Quadratic Polynomial

Quadratic polynomials are polynomials with a degree of 2. We can find the zeroes of the quadratic polynomial using various methods such as, and along with 

• Factorization Method
• Using Quadratic Formula, etc.

As Quadratic Polynomial has the highest degree of 2, there exist 2 zeros of the quadratic polynomial.

The relationship between the zeros of the quadratic polynomial and the coefficient of the quadratic polynomial is,

For any polynomial P(x) = ax² + bx + c if the zeroes of the quadratic polynomial are α, and β then,

• Sum of the zeroes (α + β) =  – Coefficient of x / Coefficient of x² = -b/a
• Product of the zeroes (αβ) = Constant term / Coefficient of x² = c/a

Let's check the relationship between zeros and coefficients of a quadratic polynomial with the help of an example.

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 since it has its highest degree as 3, there exist three zeros of a cubic polynomial. Let's suppose the zeros of the polynomials ax³ + bx² + cx + d = 0 are p, q, and r, the relationship between the zeros and polynomials and the coefficient of the polynomial will be given as

Given Cubic Polynomials,

ax³ + bx² + cx + d = 0

which has roots x = p, q and r 

• Sum of the Zeroes (p + q+ r) = – Coefficient of x²/ coefficient of x³ = -b/a
• Sum of the product of the Zeroes (pq + qr + pr) = Coefficient of x/Coefficient of x³ = c/a
• Product of the zeroes (pqr) = – Constant Term/Coefficient of x³ = -d/a

Read More

• Solving Cubic Equations
• Algebra Formulas
• Algebraic Identities of Polynomials

Examples of Relationship Between Zeroes and Coefficients of Polynomials

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