The Ungerecht Family Tree

start with quadratic equation: ax² + bx + c = 0

go to: a(x+d)² + e = 0,
  where d = b/(2a) and e = c - b²/(4a)
steps: divide by a to get x² by itself
  x² + (b/a)x + (c/a) = 0
add & subtract ((b/a)/2)², now you have:
  x² + (b/a)x + ((b/a)/2)² + (c/a) - ((b/a)/2)² = 0
  (x + (b/a))² + c/a - ((b/a)/2)² = 0
  \==========/
    quadratic   + remainder = 0
 -b/a is the vertex (x value) if graphing, y = remainder

which leads to quadratic formula: x = (-b +- √(b²-4ac)) / 2a

Multiplicity is the number of times a given factor appears in the factored form of the equation of a polynomial.

A factor with an even multiplicity will touch, odd multiplicity will cross.

 Example:
   f(x) = (x+3)(x-2)²(x+1)³
   x = -3 has multiplicity of 1, crosses the x axis like a line
   x = 2 has multiplicity of 2, touches the x axis like a parabola
   x = -1 has multiplicity of 3, crosses the x axis in s shape (half-flipped parabola)
   This has 3 different polynomials for x, and will touch or cross the x axis 3 times

What are all the possible rational roots of the following function: 6x⁴−x³−4x²−x−2

Unpacking this: first need synthetic division for polynomials: https://courses.lumenlearning.com/waymakercollegealgebra/chapter/synthetic-division/

Remainder theorem: If a polynomial f(x) is divided by x – k, then the remainder is the value f(k).

⇒ f(x)/(x-k) = f(k), or f(x)/f(k) = x-k

Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial, but first we need a pool of rational numbers to test. The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial.

The Rational Zero Theorem states that if the polynomial f(x)=anx^n+an−1x^n−1+…+a1x+a0 (coefficients are a0, a1, .. an) has integer coefficients, then every rational zero of f(x) has the form pq where p is a factor of the constant term a0 and q is a factor of the leading coefficient an. =⇒ So dividing all the p’s by all the q’s give all the possible rational zeros.

When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.