Polynomial Inequalities (DP IB Analysis & Approaches (AA): HL): 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 $x$

    • e.g. $x^4-x^2>x+6$

• The following are not polynomial inequalities

  • $\sqrt{x}>4$

  • $\frac{1}{x}\ge 3x$

    • $x^{\frac{1}{2}}$ and $x^{-1}$ 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: $P\left(x\right)\le 0$

• STEP 2
  Find the roots of the polynomial

  • You can do this by

    • factorising

    • or using your GDC to solve $P\left(x\right)=0$

• STEP 3
  Choose one of the following methods:

• Graph method

  • Sketch a graph of the polynomial

    • with or without a GDC

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

    • For $P\left(x\right)\le 0$ you want sections below the $x$-axis

    • For $P\left(x\right)\ge 0$ you want sections above 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

    • e.g. for two roots $a$ and $b$ the intervals are $x<a$, $a<x<b$ and $x>b$

  • Choose a numerical value from each interval

  • Substitute the numerical values into the original inequality (e.g. $P\left(x\right)\le 0$ or $P\left(x\right)\ge 0$) 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 $\left(x-2\right)\left(x-4\right)\left(x-6\right)>0$ the intervals are $x<2$, $2<x<4$, $4<x<6$ and $x>6$

  • For $x<2$, the expression $\left(x-2\right)\left(x-4\right)\left(x-6\right)$ is

    • negative × negative × negative = negative

    • so $\left(x-2\right)\left(x-4\right)\left(x-6\right)>0$ is not true for $x<2$

  • Repeat this for the other intervals

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