Key Points

A polynomial is an algebraic expression where the variables have only non-negative integer exponents. For example, $5x^3 - 4x^2 + x$ is a polynomial, but $\sqrt{x} + 2$ and $\frac{1}{x-1}$ are not.

The degree of a polynomial is the highest power of its variable. A polynomial of degree 1 is called linear, degree 2 is quadratic, and degree 3 is cubic.

The value of a polynomial $p(x)$ at $x=k$ is denoted by $p(k)$. A real number $k$ is called a zero of the polynomial $p(x)$ if $p(k) = 0$.

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.

A polynomial of degree $n$ can have at most $n$ real zeroes. This means a quadratic polynomial can have at most 2 zeroes, and a cubic polynomial can have at most 3 zeroes.

The graph of a quadratic polynomial $ax^2 + bx + c$ is a parabola. If the leading coefficient $a > 0$, the parabola opens upwards ($\cup$), and if $a < 0$, it opens downwards ($\cap$).

For a quadratic polynomial $ax^2 + bx + c$, if the zeroes are $\alpha$ and $\beta$, their sum is $\alpha + \beta = -\frac{b}{a}$. This is equal to $\frac{-(\text{Coefficient of } x)}{\text{Coefficient of } x^2}$.

For a quadratic polynomial $ax^2 + bx + c$, if the zeroes are $\alpha$ and $\beta$, their product is $\alpha\beta = \frac{c}{a}$. This is equal to $\frac{\text{Constant term}}{\text{Coefficient of } x^2}$.

A quadratic polynomial with given zeroes $\alpha$ and $\beta$ can be written as $k[x^2 - (\alpha + \beta)x + \alpha\beta]$, where $k$ is any non-zero constant, $(\alpha + \beta)$ is the sum of zeroes, and $\alpha\beta$ is the product of zeroes.

For a cubic polynomial $ax^3 + bx^2 + cx + d$, if the zeroes are $\alpha$, $\beta$, and $\gamma$, their sum is $\alpha + \beta + \gamma = -\frac{b}{a}$.

For a cubic polynomial $ax^3 + bx^2 + cx + d$ with zeroes $\alpha$, $\beta$, and $\gamma$, the sum of the products of zeroes taken two at a time is $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$.

For a cubic polynomial $ax^3 + bx^2 + cx + d$, if the zeroes are $\alpha$, $\beta$, and $\gamma$, their product is $\alpha\beta\gamma = -\frac{d}{a}$.