No line with a modulus ever goes under the x-axis.
Any line that does go below the x-axis, when modulated is reflected above it
Polynomials
To find unknowns in a given identity:
Substitute suitable values of x
OR
Equalize the given coefficients of like powers of x.
Factor Theorem: If (x – t) is a factor of the function p(x), then p(t) = 0
Remainder Theorem: If the function f(x) is divided by (x – t), then the remainder, R = f(t)
Binomial Series
Expanding (1 + x)^{n}, where |x| < 1
Factor Case: If the constant is not 1, pull out a factor from the brackets to make it 1, and use the general equation. Do NOT forget the indices.
Substitution Case: If the bracket contains more than one x term (e.g. 2 – x + x^{2}), then make the last part u, expand and then substitute back in.
Finding the Limit of x in the Expansion: E.g. (1 + ax)^{n}, the limit can be found by substituting ax between the modulus sign in |x| < 1 and altering it to have only x in the modulus.
Partial Fractions
Multiply through by (px + q), substitute x = ^{-q}/_{p} and find A
Multiply through by (rx + s), substitute x = ^{-s}/_{r} and find B
Multiply through by (px + q), substitute x = ^{-q}/_{p} and find A
Multiply through by (rx + s)^{2}, substitute x = ^{-s}/_{r} and find C
Substitute any constant, e.g. x = 0, and find B.
Multiply through by (px + q), substitute x = ^{-q}/_{p} and find A
Take ^{A}/_{px + q} to the other side, subtract and simplify.
The linear equation left at the top is equal to Bx + C
Improper Fraction case
If the numerator has x to the degree of power equivalent or greater than the denominator, then another constant is present.
This can be found by dividing the denominator by the numerator and using the remainder.
Example
Express the following in partial fractions:
Expand the brackets:
The greatest power of x is the same for the numerator and denominator, thus this is an improper case.
Make it into a proper fraction:
This is then written as:
Now proceed with the normal case for the fraction: