Cheat sheet · Class 9 · CBSE Mathematics

Polynomials cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

CBSE Class 9 · Mathematics

Cheat sheet

Chapter 2Polynomials

Factor Theorem: x − a is a factor if p(a) = 0.

Polynomial: $a_nx^n + \cdots + a_1x + a_0$ with whole-number powers.

By terms	By degree
Monomial – 1 term (3x)	Constant – 0 (e.g. 5); zero polynomial – not defined
Binomial – 2 terms (x + 1)	Linear – 1; Quadratic – 2
Trinomial – 3 terms	Cubic – 3

Zero of $p(x)$ : value $c$ where $p(c) = 0$ . A linear polynomial $ax + b$ has one zero, $-\dfrac{b}{a}$ .

Remainder Theorem (not in rationalised syllabus, useful): remainder when $p(x)$ is divided by $x - a$ is $p(a)$ .

Factor Theorem: $x - a$ is a factor of $p(x)$ ⇔ $p(a) = 0$ .

$x^3 - 23x^2 + 142x - 120$ : $p(1) = 0$ ⇒ $(x - 1)$ is a factor → $(x - 1)(x - 10)(x - 12)$

$ax^2 + bx + c$ : find $p, q$ with $p + q = b$ , $pq = ac$ . e.g. $6x^2 + 17x + 5 = (2x + 5)(3x + 1)$

(x + y)^2 = x^2 + 2xy + y^2

(x - y)^2 = x^2 - 2xy + y^2

x^2 - y^2 = (x + y)(x - y)

(x + a)(x + b) = x^2 + (a + b)x + ab

(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx

(x \pm y)^3 = x^3 \pm y^3 \pm 3xy(x \pm y)

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)

So if $x + y + z = 0$ , then $x^3 + y^3 + z^3 = 3xyz$ .

Remember: Factor Theorem | Split the middle term | Learn the 8 identities

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