Polynomials

Algebra is generous; she often gives more than is asked of her. — Jean le Rond d'Alembert

2.1 Introduction

A polynomial in one variable $x$ is an expression of the form

$a_n x^n + a_{n-1}x^{n-1}+\cdots+a_1 x + a_0$

with $a_i\in\mathbb{R}$, $a_n\mathrel{\htmlClass{course-neq}{\char"2260}} 0$. The degree is $n$.

A real number $k$ is a zero (or root) of $p(x)$ if $p(k)=0$.

2.2 Geometrical Meaning of Zeroes

The graph of $y=p(x)$ is a smooth curve. Zeroes are the $x$-coordinates where the graph meets the $x$-axis.

• A linear polynomial has one zero
• A quadratic polynomial has at most two zeroes
• A cubic polynomial has at most three zeroes

2.3 Zeroes and Coefficients (Quadratic)

For $p(x)=ax^2+bx+c$ with $a\mathrel{\htmlClass{course-neq}{\char"2260}} 0$, if $\alpha$ and $\beta$ are zeroes:

$\alpha+\beta=-\frac{b}{a},\qquad \alpha\beta=\frac{c}{a}$

If only one zero $\alpha$ is known (repeated root), use the same relations.

Chapter Summary

• Zero of $p(x)$: $p(k)=0$
• Graph cuts $x$-axis at zeroes
• For $ax^2+bx+c$: sum of zeroes $=-b/a$, product $=c/a$

Exercises (NCERT)

• Exercise 2.1 — geometrical meaning
• Exercise 2.2 — zeroes and coefficients
• Miscellaneous Exercise