2.8.2 Polynomial Inequalities

How do I solve polynomial inequalities?

• STEP 1: Rearrange the inequality so that one of the sides is equal to zero
  • For example: P(x) ≤ 0
• STEP 2: Find the roots of the polynomial
  • You can do this by factorising or using GDC to solve P(x) = 0
• STEP 3: Choose one of the following methods:
• Graph method
  • Sketch a graph of the polynomial (with or without a GDC)
  • Choose the intervals for x corresponding to the sections of the graph that satisfy the inequality
    • For example: for P(x) ≤ 0 you would want the sections below the x-axis
• Sign table method
  • If you are unsure how to sketch a polynomial graph then this method is best
  • Split the real numbers into the possible intervals using the roots
    • If the roots are a and b then the intervals would be x<a, a<x<b, x>b
  • Test a value from each interval using the inequality
    • Choose a value within an interval and substitute into P(x) to determine if it is positive or negative
  • Alternatively if the polynomial is factorised you can determine the sign of each factor in each interval
    • An odd number of negative factors in an interval will mean the polynomial is negative on that interval
  • If the value satisfies the inequality then that interval is part of the solution