Determine the maximum number of roots possible (the highest exponent)
The example below has at most 4 zeros.
Use the Rational Zero Theorem to list all possible rational roots (in order)
Descartes' Rule of Signs
Use Decartes' Rule of Signs to determine how many positive and negative real roots the equation may have. (Eliminate any possibilities that were found in step 2).
Make sure the polynomial is arranged in descending order.
The number of positive roots of P(x)=0 is equal to the number of changes in sign in P(x) or is less than that number by an even counting number. The number of negative roots of P(x)=0 is equal to the number of changes in sign in P(-x) or is less than that number by an even counting number.
Synthetic Division
Use synthetic division to identify the upper and lower bounds.
Upper bound - if all the sums are positive (0 counts as either).
Lower bound - if all the sums alternate in sign (0 counts as either).
Continue using synthetic division until the following occurs:
- You find a zero (remainder = 0)
- You have found the maximum number or positive or negative roots
- If c is a root, then one factor is x-c. The second factor is the quotient obtained from the synthetic division.
A. If the second factor is quadratic, solve by:
- factoring
- quadratic formula
- square root method
B. If the second factor is of degree 3 or greater, repeat steps 5-7, checking only possible roots that have not been tested earlier.
NOTE: If when using synthetic division, the remainder, P(a) and P(b), changes signs, there is at least one real zero between P(a) and P(b).