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.
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.
Table of Content
- Zeroes of a Polynomial
- Coefficients of a Polynomial
- Relationship between Zeros and Coefficients of a Polynomial
- Linear Polynomial
- Quadratic Polynomial
- Cubic Polynomial