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