Polynomials are algebraic expressions with variables and coefficients. The zeroes of a polynomial are the values that make it equal to zero, and they have a specific relationship with its coefficients.
- Zeros of a Polynomial: Values of the variable for which the polynomial becomes equal to zero.
- Coefficients of a Polynomial: Numerical values that are multiplied with the variables in a polynomial expression.
The relationship between the zeroes and coefficients of a polynomial shows how the roots of a polynomial are connected to its coefficients.
- According to the Factor Theorem, if k is a zero of P(x), then (x − k) is a factor of the polynomial.
- Using this idea, formulas are derived that relate the sum and product of zeroes to the coefficients of the polynomial.
Linear Polynomial
A linear polynomial is a polynomial of degree 1 and is written in the form ax + b, where x is a variable and a and b are constants (a ≠ 0). If P(x) = ax + b, then its zero is -b/a.
- Zero of a linear polynomial = − (constant term) / (coefficient of x)
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree 2 and is written in the form ax² + bx + c, where x is a variable and a, b, c are constants with a ≠ 0. Let α and β be the zeroes of the polynomial.
- Sum of zeroes (α + β) = −b/a = − (coefficient of x) / (coefficient of x²)
- Product of zeroes (αβ) = c/a = (constant term) / (coefficient of x²)
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 is written in the form ax³ + bx² + cx + d, where x is a variable and a, b, c, d are constants with a ≠ 0. Let α, β, γ be the zeroes of the polynomial.
- Sum of zeroes (α + β + γ) = −b/a = − (coefficient of x²) / (coefficient of x³)
- Sum of product of zeroes (αβ + βγ + αγ) = c/a = (coefficient of x) / (coefficient of x³)
- Product of zeroes (αβγ) = −d/a = − (constant term) / (coefficient of x³)
Solved Examples
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 = 2Product 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
= 0Product 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