2 polynomial functions
2.1 what is it
a polynomial can have constants, variables and exponents, but never division by a variable. exponents of variables in polynomial are 0, 1, 2,.. (zero or positive integer number)
\[P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0\]
2.1.1 degree
the degree of the polynomial is the largest exponent of that variable.
| name | degree | rule |
|---|---|---|
| constant | 0 | \(a\) |
| linear | 1 | \(ax+b\) |
| quadratic | 2 | \(ax^2+bx+c\) |
| qubic | 3 | \(ax^3+bx^2+cx+d\) |
| quartic | 4 | \(ax^4+bx^3+cx^2+dx+e\) |
2.1.2 general form
\[f(x)=ax^3+bx^2+cx+d\]
- easy to find y-intercept
- hard to graph
2.1.3 turning point form
\[f(x)=a(x-b)(x-c)(x-d)\]
- easy to find x-intercepts
- easy to graph
2.2 remainder theorem
if \(P(x)\) is divided by \((ax+b)\) the remainder is \(P(-\frac{b}{a})\)
2.3 factor theorem
if \(P(-\frac{b}{a})=0\) then \((ax+b)\) is a factor
2.4 formulas
2.4.1 quadratic formula
\[\frac{-b\pm{}\sqrt{b^2-4ac}}{2a}\]
\(f(x)=ax^2+bx+c\)
if \(f(x)=0\) then \(x=\frac{-b\pm{}\sqrt{b^2-4ac}}{2a}\)
the discriminant \(\Delta=b^2-4ac\)
- if \(\Delta>0\) then the quadratic has 2 real solutions
- if \(\Delta=0\) then the quadratic has 1 real solution
- if \(\Delta<{}0\) then the quadratic has no real solutions
the vertex is at \((-\frac{b}{2a},f(-\frac{b}{2a}))\)
2.4.2 difference of squares
\[x^2-a^2=(x+a)(x-a)\]
2.4.3 difference and sum of two cubes
\[x^3-a^3=(x-a)(x^2+ax+a^2)\] \[x^3+a^3=(x+a)(x^2-ax+a^2)\]