Polynomials

Quick Revision Polynomials

1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

2. A quadratic polynomial in $x$ with real coefficients is of the form $a x^{2}+b x+c,$ where $a, b$, $c$ are real numbers with $a \neq 0$

3. 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.

4. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.

5. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $a x^{2}+b x+c,$ then

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

6. If $\alpha, \beta, \gamma$ are the zeroes of the cubic polynomial $a x^{3}+b x^{2}+c x+d=0,$ then

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

and $\alpha \beta \gamma=\frac{-d}{a}$

7. 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 $\quad r(x)=0$ or degree $r(x)<$ degree $g(x)$