Sum and Product of zeros in a Quadratic Polynomial

Zeros of a quadratic polynomial are the values of x where the polynomial equals zero. Every quadratic polynomial has exactly two (real or complex) zeros. The sum and product of a quadratic polynomial's zeros are connected to the values of its coefficients. 
 

An algebraic expression composed of variables and coefficients is a polynomial. The general form of a polynomial with one variable x of degree n is: 
f(x) = a0xn + a1xn - 1 + a2xn - 2 +… + an - 1x + an. Where a0, a1, a2, …, an are the coefficients and a0 ≠ 0. Quadratic polynomials refer to those polynomials where the highest degree is 2. Its general form is ax2 + bx + c = 0.  
 

In a quadratic polynomial, we substitute the variable with a number to make the expression equal to zero. That particular number or value that equates the polynomial to zero is called the zeros of a quadratic polynomial. They are also called the roots of the quadratic equation. For example, consider the linear polynomial f(x) = 2x - 4, here, the value of x = 2 because f(2) = 2 × 2 - 4 = 0. 
If we consider the quadratic polynomial f(x) = x^2 - 4x + 3, the zeros are x = 1 and x = 3, since it can be factored as (x - 1)(x - 3). 
A polynomial of degree n can have up to n zeros. So, a quadratic polynomial (degree 2) can have up to two zeros, which may be real or complex. 
 

The root of a quadratic polynomial can be easily calculated using the quadratic polynomial formula. For any quadratic polynomial in the form f(x): ax2 + bx + c = 0, where a ≠ 0, the quadratic formula is: x = -b ± b2 - 4ac2a. Where a and b are the coefficients of x2 and x, respectively, and c is the constant term. 
 

For a quadratic polynomial in the form ax2 + bx + c, the two roots (say α and β) represent the values of the variable (x) that make the polynomial equal to zero. Here, the sum of the roots is: α + β = -b/a. 
For example, f(x) = 3x2 - 5x + 2 
Here, a = 3, b = -5, and c = 2
α + β = -b/a
= -(-5)/3 = 5/3
So, the sum of the roots of 3x2 - 5x + 2 = 5/3. 
 

Let the roots of a quadratic polynomial f(x) = ax2 + bx + c be α and β. Then their product can be calculated using the formula: αβ = c/a.
For example, f(x) = 3x2 - 5x + 2
The product of roots of the quadratic polynomial is: αβ = c/a
Here, c = 2 and a = 3
αβ = 2/3
 

In this section, we will discuss the relation between the coefficients and the sum and product of the zeros of the polynomial. Let the roots of a quadratic polynomial in the form f(x) = ax2 + bx + c be α and β. If the roots are α and β, then the sum is: α + β = -b/a. Their product is: αβ = c/a.

For example, finding the sum and product of zeros in f(x) = 2x2 + 5x + 3
Here, a = 2, b = 5, and c = 3.

Using the formula to find the sum and product of the roots:
α + β = -b/a and αβ = c/a
α + β = -5/2 
αβ = 3/2

Let’s verify the answer by finding the roots of the quadratic polynomial. To find the roots, we use the quadratic formula:
x = -b ± b2 - 4ac2a
x = -5 ± 52 - 4 × 2 × 32 × 2
x = -5 ± 25 - (4 × 2 × 3)4
x = -5 ± 25 - 244
x = -5 ± 14 
x = -5 +  14  or x = -5 -  14 
x = -4/4 = -1 or
x = -6/4 = -3/2 
So, x = -1 and x = -3/2
The sum of the roots = -1 + (-3/2) = -5/2 
The product of the roots = -1 × (-3/2) = 3/2
Therefore, the sum and product of the roots are directly related to the coefficients. 

Let’s understand how the sum and product of a quadratic polynomial’s roots are related to its coefficients. The factored form of a quadratic polynomial is: f(x) =  (x - a)(x - b), where a and b are the roots. Let the sum of the roots be S = a + b and the product be P = ab. 
Expanding the equation: (x - a)(x - b) = x2 - ax - bx + ab
= x2 - (a + b)x + ab
So, P(x) = x2 - Sx + P. 
Here, the coefficient of x is -S, so it is the negative of the sum of the roots and the term P is the product of the roots. 

Now, let’s generalize the result for any quadratic polynomial of the form: f(x) = ax2 + bx + c. where ‘a’ is the coefficient of x2, b is the coefficient of x, and c is the constant. Consider the root as m and n, then the polynomial can be factorized as: f(x) = a(x - m)(x - n)

Let’s compare two forms of a quadratic polynomial:  
ax2 + bx + c and a(x - m)(x - n)
Now divide both sides by ‘a’ to simplify:
x2 + (b/a)x + c/a = (x - m)(x - n)
Next, expand the right-hand side: (x - m)(x - n) = x2 - (m + n)x + mn
So now we have:
x2 + (b/a)x + c/a = x2 - (m + n)x + mn
Since both expressions represent the same polynomial, we can match the coefficients:
The coefficient of x: 
b/a = -(m + n)  m + n = -b/a
The constant term:
c/a = mn
So we’ve shown:
Sum of the roots: m + n = -b/a
 

The sum and product of zeros in a quadratic polynomial 

• To predict the motion of an object like a basketball or a rocket in projectile motion, we use the quadratic formula, as the root of the quadratic formula gives the time when the object hits the ground. The sum and product of the roots help us understand the total time taken and how the time intervals are connected. 
• To find the profit maximization, we can use the sum and product of zeros in a quadratic polynomial to find the best price to sell products. As the sum of zeros gives the average number of items sold, and the product of zeros helps in finding the range of sales to avoid losses. 
• In video games, to make the movements like jumping look smooth and realistic with the help of the sum and product of zeros are used. Here, the total jump distance is calculated using the sum of zeros, and the starting point is decided using the product of zeros.