Rational Zero theorem

If P(x)= the anxn+an-1 xn-1 dah dah dah a1x a0 has integer coefficients (an cannot be zero)

and p/q is a rational zero in lowest terms of P then


p is a factor of the constant term and

q is a factor of the leading coefficient

Guidelines for FInding the zeros of a polynomial function with integer coefficients

1. Gather general information. Determine the degree N of the polynomial function. The number of the distinct zeros of the polynomial function is at most N
Apply Descartes' Rule of Signs to find the possible number of positive zeros and also the possible number of negative zeros

2. Check suspects. Apply the Rational Zero Theorem to list rational numbers that are possible zeros. Use syn div to test the numbers in your list. If you find an upper or lower bound then eliminate from your list any number that is greater than the upper bound or less than the lower bound

3 Work with reduced polynomials. each time a s zero is found, you obtain a reduced polynomial

if a reduced polynomial is of degree 2, find its zeros either by factoring or by applying the quadratic formula.

if the degree of a reduced polynomial is 3 or greater repeat the above steps for this polynomial.

Descartes' Rule of Signs

Let P be be a polynomial function with real coefficients and with the terms arranged in order of decreasing powers of x

1. The number of positive real zeros of P is equal to the number of variations in the sign of P(x), or to that number decreased b an even integer.

2. The number of negative real zeros of P is equal to t he n umber of variations in sign of P(-x), or to that number decreased by an even integer.

Properties of Exponential and Logarithimic functions

Quiz 4

Exponential Functions

f(x)=b to the x power

where b>0

b cannot = 1

b is the base


x is any real number

Inverse Functions

To find the inverse of a function you interchange x for y or in other words (6,0) = (0,6) in inverse

essentially what you are doing is taking the domain y and switching it with the range x therefore an inverse function has some of these properties

The domain of f(x) = f(x)^-1 range and vice versa

this causes the x intercepts and y intercepts to be switched along with Horizontal and Vertical asymptotes to be switched.