Polynomials

Long division
Synthetic division
- Dividing a polynomial by a linear factor
- e.g. Divide $2 x^{3} - 3 x^{2} + 2 x - 5$ by $x - 2$ ; while if by $2 x - 1$ , we use the 1/2 in the synthetic division, but in the final result, we factor 2 to multiply $(x - \frac{1}{2})$
Remainder theorem
- When a polynomial P(x) is divided by x-a, the remainder is P(a)
Factor theorem
- Let P(x) be a polynomial, (x-a) is a factor of P(x) if and only if P(a) = 0
P= QD + R
- polynomial, quotient, divisor, remainder
Rational roots theorem
- Possible rational roots roots = $\pm \frac{f a c t o r s o f c o n s t a n t t e r m}{f a c t o r s o f l e a d i n g c o - e f f i c i e n t}$ .
Factoring / Solving polynomials
- $y = a_{n} x^{n} + a_{n - 1} x^{n - 2} + . . . . . . + a_{2} x^{2} + a_{1} x + a_{0}$
- where $a_{n}$ $\neq$ 0, $a_{n}, a_{n - 1}$ ... are real numbers, and n $\in$ W
whole number (W): integers (Z) no less than 0
- etc. 0,1,2,3...
Look at a graph and answer questions
- $y = a (x - p)^{m} (x - q)^{n} (x - r)^{l} . . . . . .$
zeros/roots, x-int
- usually the testing root would be at the low end of the possible roots
y-int when x=0
leading co-efficient determines end behavior on the right
- a > 0, graph rises to the right
- a < 0, graph falls to the right
Degree determines the general shape
Turning points at each x-int using multiplicity of factored polynomials

Graphing

find the x-int
find the y-int
the end behavior using the leading co-efficient
the shape of the polynomial function using the degree and the end behavior
turning points using multiplicity
- when the multiplicity = 1, the graph crosses the x-axis at its root
- tangent point - multiplicity = 2n (n $\geq$ 1)
  - touch and turn
- inflection point - multiplicity = 2n+1 (n $\geq$ 1)
  - counted as 2 turning points
  - touch and cross
- If a polynomial has n turning points, it's of minimum degree of n+1
use x-values between the x-int with a table of x, y values to draw a smooth curve

x-intercept
- per Joanna, x-int would be the point coordinates where a line crosses the x-axis
- per workbook, x-int: 2, -3, etc.
- to indicate when graphing, can indicate the point
y-intercept
- The point where the line crosses the y-axis
- seems not using y-int=-2

1/x not considered a polynomial
Discriminant: $b^{2} - 4 a c$
every real number is complex number, a complex number is not necessarily a real number, e.g. $i$ , $π - i$ ...
table of values needed for transformation sometimes as verification