A single-variable polynomial is an equation of the form:
\(\sum^n_{i=0} a_i x^i=0\)
For example:
\(x=1\)
\(x^2=4\)
\(x^2-3x+2=0\)
The degree of a polynomial is the highest-order term.
For example \(x^3+x=0\) has degree \(3\).
A solution to a polynomial is a root.
For example \(1\) and \(2\) are roots of \(x^2-3x+2=0\)
Quadratic polynomials are of the form \(ax^2+bx+c=0\).
\(x=\dfrac{-b\pm \sqrt {b^2-4ac}}{2a}\)
We can get the two solutions to a quadratic equation from the following manipulation.
\(ax^2+bx+c=0\)
\(a[x^2+\dfrac{b}{a}x]=-c\)
\(a[(x+\dfrac{b}{2a})^2-\dfrac{b^2}{4a^2}]=-c\)
\(a[(x+\dfrac{b}{2a})^2]=\dfrac{b^2}{4a}-c\)
\((x+\dfrac{b}{2a})^2=\dfrac{b^2-4ac}{4a^2}\)
\(x+\dfrac{b}{2a}=\pm \sqrt {\dfrac{b^2-4ac}{4a^2}}\)
\(x=\dfrac{-b\pm \sqrt {b^2-4ac}}{2a}\)