Polynomial Inequalities (DP IB Analysis & Approaches (AA)): Revision Note

Polynomial inequalities

What is a polynomial inequality?

• A polynomial inequality is an inequality where both sides are polynomials

  • i.e. have constant terms or positive integer powers of

    • e.g.

• The following are not polynomial inequalities

  • 

  • 

    • and are not positive integer powers

How do I solve polynomial inequalities?

• STEP 1
  Rearrange the inequality so that one of the sides is equal to zero

  • For example:

• STEP 2
  Find the roots of the polynomial

  • You can do this by

    • factorising

    • or using your GDC to solve

• STEP 3
  Choose one of the following methods:

• Graph method

  • Sketch a graph of the polynomial

    • with or without a GDC

  • Choose the intervals (ranges) of corresponding to the sections of the graph that satisfy the inequality

    • For you want sections below the -axis

    • For you want sections above the -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

    • e.g. for two roots and the intervals are , and

  • Choose a numerical value from each interval

  • Substitute the numerical values into the original inequality (e.g. or ) to see if it is true

    • The solutions are the intervals in which it is true

• Alternatively if the polynomial is factorised you can determine the sign of each factor in each interval

  • e.g. for the intervals are , , and

  • For , the expression is

    • negative × negative × negative = negative

    • so is not true for

  • Repeat this for the other intervals

    • The solution will be the intervals in which it is true