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CAPE Pure Math: The Module 1 (Unit 1) Formula Sheet

Just a quick summary of the useful formulae and info from module 1. (In progress)


Remainder and Factor Theorem

Where a is the value of x when a divisor (x-a) is equated to zero and R is the remainder when f(x) is divided by x-a. f(x) is any polynomial.


Factors of a^n-b^n


Cubic Roots

Where alpha, beta and gamma are the roots of the cubic equation.


Problems involving cubic roots related to equations usually take the form 'find the equation with roots...' So, here are two common forms to remember:


Summation notation for fancy boys and sigma males (sorry)


If the new roots are the squares of the previous roots, the sum and sum of the products of the roots could be found as shown above.


If the new roots are the inverses of the previous roots:


Sigma Notation/Summation






Modulus/Absolute Value Function

The definition of the modulus function:

The results of the modulus function:


Inequalities involving the modulus function:


The triangle inequality proof:

Rational Functions

(Why are they here? Why not!)


A rational function is a function of the form:


The domain of a rational function is all values of x for which the denominator (Q(x)) is not zero.


Asymptotes


Vertical- The vertical asymptote occurs where the denominator is zero, i.e. equate the denominator to zero and solve for x.


Horizontal- Depends upon the degree of the numerator (n) and denominator (m)

  1. If n>m, there is no horizontal asymptote. However, if n=m+1, there is an oblique asymptote, where the asymptote's equation is the quotient of the denominator and numerator.

  2. If n<m, then the x-axis is the horizontal asymptote.

  3. If n=m, then the horizontal asymptote exists at

X-intercept- equate the numerator to zero and solve for x.


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