What is a Polynomial?
Polynomial is an algebraic expression, in which the variables have non-negative powers.
Types of Polynomial
Linear Polynomial
A polynomial of degree 1 is called a linear polynomial.
For example, ax – 3 , $a \neq 0$
Quadratic Polynomial
A polynomial of degree 2 is called a quadratic polynomial.
For example, $ax^2+ bx + c $, $a \neq 0$
Cubic Polynomial
A polynomial of degree 3 is called a cubic polynomial.
For example, $ax^3+ bx^2 + cx+ d $, $a \neq 0$
Polynomial Formulas
Exercise 2.1
Graphical method to find zeroes:
The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x -axis.
Total number of zeroes in any polynomial equation = total number of times the curve intersects x-axis.
Exercise 2.2
Quadratic Polynomial
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $ax^2$ + bx + c, then
Sum of zeroes = $\alpha + \beta = \dfrac{-b}{a}$ = $\dfrac{-(Coefficient\:\:of\:\:x)}{(Coefficient \:\:of \:\:x^{2})}$
Product of zeroes = $\alpha \times \beta $ =$\dfrac{c}{a}$ = $\dfrac{Constant \:term}{(Coefficient \:\: of \:\: x^{2})}$
Cubic Polynomial
If $\alpha, \beta, \gamma$ are the zeroes of the cubic polynomial $ax^3$ + bx2 + cx + d, then
$\alpha + \beta + \gamma = \dfrac{-b}{a}$
$\alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}$
$\alpha \times \beta \times \gamma = \dfrac{-d}{a}$
Exercise 2.3
The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that
p(x) = g(x) q(x) + r(x),
where r(x) = 0 or degree r(x) < degree g(x)