Sum and Product of Zeroes in a Quadratic Polynomial

There are always two zeros for any quadratic polynomial. The zeros of a polynomial are those values of the variable for which the polynomial as a whole has zero value. The sum and product of zeros in a quadratic polynomial have a direct relation with the coefficients of variables in the polynomial. Then it becomes very easy to calculate the sum and product of zeros even without knowing the values of zeros of a polynomial. Let's learn about it in detail in this lesson.

Polynomials are expressions formed by variable(s). A polynomial in one variable x of degree n is of the form: f(x) = a₀xⁿ + a₁x^{n - 1} + a₂x^{n - 2} + ... + a_{n - 1}x + a_n, where a₀, a₁,.....a_n-1, a_n are coefficient of the n terms of the polynomial and a₀ ≠ 0.

A quadratic polynomial is an expression with the highest degree 2. It's a degree two polynomial as the highest power of variables is 2.

The zeros of a polynomial are those values of the variable for which the polynomial as a whole has zero value. For example, consider the linear polynomial: p(x) = x+2. The zero of the polynomial is -2 because when x = -2, p(x)=0. Consider a quadratic polynomial: p(x)= x²−9. The zeros of the polynomial are ±3. When x = ±3, p(x)=3²−9=0.

Note that when the coefficient of all the terms and the constant of the polynomial is 0, the polynomial is called a zero polynomial and is denoted by 0.

A polynomial of degree n will have n number of zeros or roots. Thus, a quadratic polynomial can have at most two zeros, whereas a cubic polynomial can have at most 3 zeros.

Quadratic polynomial formula helps us to solve the value of a polynomial with degree 2 easily and quickly. A quadratic polynomial is of the form p(x): ax²+bx+c, where a≠0.

Note: "Quad" means 2. The degree of a polynomial has to be 2. Hence, a which is the coefficient of x² cannot be 0. The quadratic formula to find the solution to a quadratic equation is given by: \(\begin{align}x =\frac{-b \pm \sqrt{b^2-4ac}}{2a}\end{align}\), in which,

• a is the coefficient of x² and is also called the quadratic coefficient
• b is the coefficient of x and is also called the linear coefficient and
• c is the constant term

As you might know that there are two roots of a quadratic polynomial. If α and β are the zeros of the quadratic polynomial f(x): ax²+bx+c, the sum of the roots of the polynomial is: α+β= −b/a. In other words, it refers to (-coefficient of x)/ (coefficient of x²).

If α and β are the zeros of the quadratic polynomial f(x): ax²+bx+c, the product of the roots of the polynomial is: αβ= c/a.

In other words, it refers to (constant term)/ (coefficient of x²).

Let's summarize the sum and product of the roots formula of a quadratic equation through the image below:

In this section, we will explore the relationship between the coefficient and the sum and product of the zeros of any quadratic polynomial. Now that we have seen the crucial role played by the coefficients in determining the characteristics of any polynomial, we turn our attention to a special case: quadratic polynomials. In particular, we want to see how the zeroes of a quadratic polynomial are related to its coefficients. To understand this relation between the coefficient and the zeros, let’s consider some examples.

Let, a(x): x²−3x+2. The coefficient of x in this polynomial is –3, while the constant term is 2. To determine the zeroes of this polynomial, we factorize it: a(x)= (x−1)(x−2), where Zeroes are x=1 and x=2. Note that the sum of the zeroes is 3, which is the negative of the coefficient of x in the polynomial, while the product of the zeroes is 2, the same as the constant term. Is there some relation here?

Let’s take another example. Consider b(x): x²−7x+6. The zeroes can be determined by factorizing: b(x)= (x−1)(x−6)  ⇒  Zeroes are, x=1, x=6. The sum of the zeroes is 7, once again the negative of the coefficient of x, while their product is 6, equal to the constant term of the polynomial. This cannot be a coincidence, so let us explore this more formally. Consider the following quadratic polynomial p(x): (x−a)(x−b). The zeroes are a and b. Let us denote the sum and product of the zeroes by S and P respectively. S= a+b and P= ab.

Next, we note that the polynomial can be written (by multiplication) as follows: p(x)= x²−ax−bx+ab ⇒ p(x)= x²−(a+b)x+ab ⇒ p(x)=x²−Sx+P

Now the relation should be clear!

The coefficient of x in the polynomial is the negative of the sum of its roots, while the constant term is the same as the product of the roots. Note that we assumed the polynomial p to be of the form p(x): (x−a)(x−b). That is, the coefficient of the square term in this polynomial is 1. However, in an arbitrary quadratic polynomial, this coefficient can be any non-zero real number. Therefore, let us generalize our result.

Consider the following quadratic polynomial p(x): lx²+mx+n. The coefficient of the square term is l. Let the zeroes of this polynomial be a and b. Then, we can also write it as follows (note this very carefully!), p(x): l(x−a)(x−b). Make sure you understand this.

Now, we compare the two expressions (corresponding to the same polynomial):

\[\begin{align}&l{x^2} + mx + n = l\left( {x - a} \right)\left( {x - b} \right)\\&\Rightarrow \;\;\;{x^2} + \frac{m}{l}x + \frac{n}{l} = \left( {x - a} \right)\left( {x - b} \right)\\&\Rightarrow \;\;\;{x^2} + \frac{m}{l}x + \frac{n}{l} = {x^2} - \left( {a + b} \right)x + ab\\&\Rightarrow \;\;\;{x^2} + \frac{m}{l}x + \frac{n}{l} = {x^2} - Sx + P\\&\Rightarrow \;\;\; - S = \frac{m}{l},\;P = \frac{n}{l}\\&\Rightarrow \;\;\;\left\{ \begin{gathered}S =  - \frac{m}{l} =  - \frac{{{\rm{coeff}}\;{\rm{of}}\;x}}{{{\rm{coeff}}\;{\rm{of}}\;{x^2}}}\\\!\!\!\!\!\!\!\!\!\!P = \frac{n}{l} = \frac{{{\rm{constant}}}}{{{\rm{coeff}}\;{\rm{of}}\;{x^2}}}\end{gathered} \right.\end{align}\] This is an extremely important result.

In any quadratic polynomial:

• The sum of the zeroes is equal to the negative of the coefficient of x by the coefficient of x².

• The product of the zeroes is equal to the constant term by the coefficient of x².

Knowing how we derive this result is crucial, so make sure that you fully understand it. Note a particular case. If the coefficient of x² is 1, that is, if the polynomial is of the following form p(x): x²+ex+f, then the sum and product of the zeroes are simply: S= −e/1= −e and P= f/1= f.

Important Notes:

• The quadratic formula to find the roots of a quadratic equation p(x):ax²+bx+c is \(\begin{align}{-b \pm \sqrt{b^2-4ac} \over 2a}\end{align}\).
• The sum of the zeroes of a quadratic polynomial equals the negative of the coefficient of x by the coefficient of x².
• The product of the zeroes equals the constant term by the coefficient of x².
• A polynomial having the value 0 is called a zero polynomial.

What is the Sum of the Roots of a Cubic Equation?

A cubic equation is of the form ax³+bx²+cx+d=0. The sum of the roots of the cubic equation is −b/a. That is coefficient of x²/coefficient of x³.

How to Find the Zeros of a Polynomial?

Zeros of any polynomial can be found by following the steps given below: 

• Step 1: Use the Rational Zero Theorem to list all possible rational zeros of the polynomial. Rational zero of the polynomial = constant term/ leading term.
• Step 2: Use synthetic division to evaluate all given possible zeros
• Step 3: Repeat step 2 using the quotient found with synthetic division; continue until the quotient is a quadratic polynomial.
• step 4: Use the quadratic formula to find the factors of the quadratic polynomial

What are the Zeros of a Polynomial?

The zeroes of a polynomial are those values of the variable for which the polynomial as a whole has zero value. For example, consider the polynomial p(x)= x²−16. The zeros of the polynomial are ±4. When  x= ±4, p(x)=0.

How many Zeros does a Polynomial have?

A polynomial of n degree has n zeros. 

• A linear polynomial has 1 zero.
• A quadratic polynomial has 2 zeros.
• A cubic polynomial has 3 zeros.
• A polynomial of degree 8 has 8 zeros.

What is the Sum of the Zeros of a Quadratic Polynomial?

The Sum of zeros of a quadratic polynomial given as ax²+bx+c can be found by taking the negative value of the ratio of the coefficient of x by the coefficient of x². It is given as -b/a.

How to Form Quadratic Polynomials with Sum and Product of its Zeros?

Quadratic polynomials can be written as p(x): x²- Sx + P, where S is the sum of zeros and P is the product of zeros.

How do you Find the Sum and Product of the Roots of a Quadratic Equation?

For a quadratic equation, ax²+bx+c= 0, the sum of the roots is -b/a and the product of the roots is c/a.