Unit 2 · Class 10Mathematics (Basic)

Polynomials

Zeroes of a polynomial and their relationship with coefficients; division algorithm for polynomials.

Lessons

6 concepts

1
Zeroes and Coefficients (Quadratic)
If $\alpha, \beta$ are zeroes of $ax^2 + bx + c$ ( $a\ne 0$ ), then $\alpha + \beta = -b/a$ and $\alpha\beta = c/a$ . This lets you construct a quadratic with given zeroes: $x^2 - (\alpha+\beta)x + \alpha\beta = 0$ .
2
Geometrical Meaning of Zeroes
The graph of $y = ax^2 + bx + c$ is a parabola. Its real zeroes are the $x$ -coordinates of the points where the parabola meets the $x$ -axis. A quadratic can cross the $x$ -axis in 2 points (2 real zeroes), touch it (equal…
3
Division Algorithm for Polynomials
For polynomials $p(x)$ and $g(x)$ (with $g(x) \ne 0$ ), there exist unique polynomials $q(x)$ (quotient) and $r(x)$ (remainder) with $p(x) = g(x)\, q(x) + r(x)$ and either $r(x) = 0$ or $\deg r(x) < \deg g(x)$ . This mirro…
4
Factor and Remainder Theorems
Remainder Theorem: if $p(x)$ is divided by $(x-a)$ , the remainder is $p(a)$ . Factor Theorem: $(x-a)$ is a factor of $p(x)$ iff $p(a) = 0$ . These reduce factorisation to root-hunting.
5
Cubic Polynomial Relations
For a cubic $ax^3 + bx^2 + cx + d$ with zeroes $\alpha, \beta, \gamma$ : $\alpha+\beta+\gamma = -b/a$ , $\alpha\beta+\beta\gamma+\gamma\alpha = c/a$ , $\alpha\beta\gamma = -d/a$ . Use these Viète-style identities to construc…
6
Strategy: Splitting the Middle Term
To factor $ax^2 + bx + c$ , find two numbers $m, n$ with $m+n = b$ and $mn = ac$ . Rewrite $bx$ as $mx + nx$ and factor by grouping. E.g. for $6x^2 + 11x + 3$ : need product $18$ , sum $11$ ; pick $9,2$ : $6x^2 + 9x + 2x + 3 =…

Skills to practise

Skill · medium

Polynomials — Practice