Maharashtra State Board · Class 10 · Mathematics · Chapter 2

Polynomials — Formula Sheet

Board Formulas

10 formulas

1.Standard Form of Quadratic Polynomial★
$p (x) = a x^{2} + b x + c, a \neq = 0$
$a, b, c$ : Real coefficients with a ≠ 0
Degree = 2. If a > 0 parabola opens upward; if a < 0 it opens downward.
2.Sum of Zeros (Quadratic)★
$α + β = - \frac{b}{a}$
$α, β$ : Zeros (roots) of the quadratic
Sum of the two roots equals -(coefficient of x)/(coefficient of x²).
3.Product of Zeros (Quadratic)★
$α β = \frac{c}{a}$
Product of the two roots equals (constant term)/(leading coefficient).
4.Quadratic Polynomial from Given Zeros
$p (x) = k [x^{2} - (α + β) x + α β]$
k is any non-zero constant. Take k = 1 for the simplest form.
5.Sum of Zeros (Cubic)
$α + β + γ = - \frac{b}{a}$
For p(x) = ax³ + bx² + cx + d. Same sign rule as quadratic.
6.Sum of Products Taken Two at a Time (Cubic)
$α β + β γ + γ α = \frac{c}{a}$
Middle Vieta relation for a cubic polynomial.
7.Product of Zeros (Cubic)
$α β γ = - \frac{d}{a}$
Sign is negative for cubic — differs from the quadratic formula.
8.Division Algorithm for Polynomials
$p (x) = g (x) \cdot q (x) + r (x)$
Dividend = Divisor × Quotient + Remainder, with deg r(x) < deg g(x).
9.Remainder Theorem
$p (x) divided by (x - a) \Rightarrow r = p (a)$
Substitute x = a into p(x) to get the remainder directly. Bypasses long division.
10.Factor Theorem
$(x - a) is a factor ⟺ p (a) = 0$
Trial-check integer factors of the constant term for cubic factorisation.