Factoring Polynomials

A Review Of Some Concepts (for myself):
Remainder Theorem:
When dividing a polynomial expression p(x) by a linear polynomial expression x-b, the remainder is p(b).

Factor Theorem:
A linear expression x-b is only a factor of p(x) if p(b) is 0. (So there must not be any remainder)

Integral Zero Theorem:
If x=b is an integral zero of the polynomial P(x), then the constant term of P(x) must be divisible by 2.

Rational Zero Theorem:
If x=b/a is a rational zero of the polynomial P(x), then the constant term of P(x) must be divisible by b, and the leading coefficient must be divisible by a.

Sinusoidol Functions:Transformations

This is another brief post for my exam studying.

Like most other kinds of functions, sinusoidal functions can can have transformations applied on their graphs:

f(x) = a sin (k(x-d))+ c

a is the vertical stretch and/or reflection. |a| is the amplitude.

k is the horizontal stretch and/or reflection, and it affects the period of the function. The period is |360÷k|

d is the horizontal translation. It doesn't affect the properties of the function

c is the vertical translation, and it determines what the equation of the axis is: the equation of axis is y=c

Also, using a and c, one can find the maximum and minimum points on the graph:

• maximum value= c+ |a|
• minimum value = c- |a|

Strategies for Proving Trigonometric Identities

From all the trig identity problems that I have solved, I have come up with these strategies:

1. Convert everything to sine and cosine. In other words, get rid of the tangents, cotangents, cosecants and secants.
2. Work with the more complex side first: it's often easier to simplify expressions that are complex than to make the more simple side equal to the more complicated one.
3. Use common denominators: if the two sides have different denominators, how can they equal to each other?
4. If the sines and cosines have 2 or 4 as their exponents, then try to use the identity sin² θ + cos² θ = 1 to simplify.
5. Don't just work on one side: unless one of the sides is a simple value like "1" or "sin θ", it's usually a good idea to try to make the two sides meet in the middle.
6. Factor: A lot of times it is important to factor, as this would often help simplifying the sides

Parallel Lines: The Different Pairs of Angles

Note: This is more of a review for myself...

When two parallel lines are cut by a transversal line, eight angles will be formed, like this:

In such a situation, you can identify four types of angle pairings: alternate exterior angles, alternate interior angles, corresponding angles, and same-side interior angles.