Polynomials Chapter 2 class 10 Basic Concepts

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


2. Quadratic Polynomial:
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. Zeros of a Polynomial:
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. No. of Roots:
A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.


5.Relation between Roots and Zeros of a Quadratic Polynomial:
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. Relation between Roots and Zeros of a Cubic Polynomial:
If $\alpha, \beta, \gamma$ are the zeroes of the cubic polynomial $a x^{3}+b x^{2}+c x+d$, then

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


7. Division Algorithm:
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 $g(x)$.