Relation between zeroes and coefficients of a polynomial

Relationship Between Zeroes and Coefficients of a Polynomial

1. Linear Polynomial

A linear polynomial is of the form ax + b, where a ≠ 0. The zero of a linear polynomial is the value of x that makes the polynomial equal to zero.

Zero of ax + b:
Let ax + b = 0
⇒ x = -b/a

Therefore, the root of the linear polynomial can be found by taking the negative of the constant term divided by the coefficient of x.

2. Quadratic Polynomial

A quadratic polynomial is of the form ax² + bx + c, where a ≠ 0.

Let α and β be the roots of the quadratic polynomial. In that case, the relationship between the roots and the coefficients can be expressed as:

• Sum of zeroes (α + β) = -b/a
• Product of zeroes (αβ) = c/a

These relationships are derived from the factorized form: a(x - α)(x - β) = ax² - a(α + β)x + aαβ

3. Cubic Polynomial

A cubic polynomial is of the form ax³ + bx² + cx + d, where a ≠ 0.

If α, β, and γ are the zeroes of the polynomial, then:

• Sum of zeroes (α + β + γ) = -b/a
• The total of the products of the roots taken in pairs (αβ, βγ, and γα) is equal to c/a
• Product of zeroes (αβγ) = -d/a

These relationships are derived from the expansion of a(x - α)(x - β)(x - γ).