Relationship Between Zeroes & Coefficients | Polynomials

01 · KEY FORMULAS

Relationship for Quadratic Polynomials

If α and β are the zeroes of a quadratic polynomial p(x) = ax² + bx + c (where a ≠ 0), then:

Sum of zeroes (α + β) = -b / a = - (Coefficient of x) / (Coefficient of x²)Product of zeroes (α · β) = c / a = (Constant term) / (Coefficient of x²)

✓

Forming a Quadratic Polynomial

If sum of zeroes (S) and product of zeroes (P) are given, the quadratic polynomial is:
k[x² - Sx + P] (where k is a non-zero real constant)

02 · SOLVED NCERT EXAMPLES

Step-by-Step Problem Solving

EXAMPLE 1

Find zeroes of x² + 7x + 10 and verify relationship with coefficients

Step 1: Factorise p(x):

x² + 7x + 10 = (x + 2)(x + 5)
Setting p(x) = 0 gives zeroes: α = -2 and β = -5

Step 2: Verify Sum of Zeroes:

α + β = (-2) + (-5) = -7
-b / a = -7 / 1 = -7 ✓ (Verified)

Step 3: Verify Product of Zeroes:

α · β = (-2) × (-5) = 10
c / a = 10 / 1 = 10 ✓ (Verified)

EXAMPLE 2

Find a quadratic polynomial whose sum and product of zeroes are -3 and 2

Given: Sum of zeroes (S) = α + β = -3, Product of zeroes (P) = α · β = 2.

The standard formula for polynomial is k[x² - Sx + P]:

p(x) = k[x² - (-3)x + 2]
p(x) = x² + 3x + 2 (taking k = 1)

03 · CUBIC POLYNOMIALS

Relationship for Cubic Polynomials

For a cubic polynomial p(x) = ax³ + bx² + cx + d with zeroes α, β, γ:

α + β + γ = -b / aαβ + βγ + γα = c / aα · β · γ = -d / a

04 · QUICK SUMMARY

Key Formulas

For ax² + bx + c: α + β = -b/a and α · β = c/a.

Quadratic Polynomial = k[x² - (α + β)x + αβ].

To find zeroes, factorise p(x) by splitting the middle term and set each factor to 0.