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Relationship between Zeroes and Coefficients of a Polynomial

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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 x 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. 

Intercept on X and Y axis for y = 4x+2

 

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

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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

FAQs on Relationship Between Zeroes and Coefficients of Polynomials

Q1: What does the Degree of a Polynomial mean?

Answer:

The degree of the polynomial is the highest power of the independent variable it tells us how many time the curve cuts the x-axis as linear polynomial in x cut the x-axis once, whereas the quadratic polynomial in x cut the x-axis twice, etc.

Q2: What is the Relationship Between Coefficient of Polynomials and Roots of Polynomials?

Answer:

Their is a definite relationship between coefficient of polynomials and roots of polynomials for a quadratic polynomial ax2 + bx + c = 0 if the zeroes of the polynomial are p, and q then,

  • Sum of the roots (p+q) = -b/a
  • Product of the roots(pq) = c/a

Q3: What Is the Relationship Between Zeroes and Coefficients of a Cubic Polynomial?

Answer:

For the cubic polynomial, ax3 + bx2 + cx + d =0. The relationship between zeroes and coefficients of a cubic polynomial is,

  • Sum of Zeroes = – Coefficient of x2/ coefficient of x3 =  = -b/a
  • Sum of product of Zeroes = – Coefficient of x/Coefficient of x3 = c/a 
  • Product of Zeroes =  – Constant term/Coefficient of x = -d/a

Q4: What is the Relationship Between Zeroes and Coefficients of a Linear Polynomial?

Answer: 

For the linear polynomial ax + b = 0. The relationship between zeroes and coefficients of a linear polynomial is,

Zero of Linear Polynomial = (-b/a) = (– Constant Term/Coefficient of x)


Last Updated : 17 Jun, 2023
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