CBSE · Class 10 · Mathematics · Chapter 2

Polynomials — Formula Sheet

Board Formulas

10 formulas

1.Standard Form of a Polynomial
$p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$
Highest exponent n is called the degree. Leading coefficient aₙ ≠ 0.
2.Zero of a Polynomial★
$p (α) = 0 ⟺ α is a zero of p (x)$
Zero of p(x) = root of p(x) = 0 = x-coordinate of the graph where it crosses the x-axis.
3.Zeroes of a Quadratic — Sum★
$α + β = - \frac{b}{a}$
For ax² + bx + c = 0. Note the MINUS sign.
4.Zeroes of a Quadratic — Product★
$α β = \frac{c}{a}$
For ax² + bx + c = 0. Positive if the leading and constant terms share sign.
5.Form Quadratic from Zeroes
$x^{2} - (α + β) x + α β = 0$
Any scalar multiple of this is also a quadratic with the same zeroes.
6.Cubic — Sum of Zeroes
$α + β + γ = - \frac{b}{a}$
For ax³ + bx² + cx + d = 0. Same negative sign pattern as quadratic.
7.Cubic — Sum of Products (Two at a Time)
$α β + β γ + γ α = \frac{c}{a}$
The MIDDLE symmetric function of the roots. Sign is positive.
8.Cubic — Product of Zeroes
$α β γ = - \frac{d}{a}$
Note the MINUS sign (odd degree).
9.Division Algorithm for Polynomials★
$p (x) = g (x) q (x) + r (x) with de g r (x) < de g g (x)$
Analogue of division algorithm for integers. r(x) may be zero (⇒ g(x) is a factor).
10.Factor Theorem
$(x - α) divides p (x) ⟺ p (α) = 0$
Quick way to check whether a linear factor works. Cornerstone of factorisation.