2.8.2 Polynomial Inequalities

Solving Polynomial Inequalities - Graph Method

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

  • P(x) ≤ 0

• STEP 2: Find the roots of the polynomial

  • Factorizing to solve P(x) = 0

• STEP 3: Graph method

  • Sketch a graph of the polynomial

  • Choose the intervals for x corresponding to the sections of the graph that satisfy the inequality

    • For P(x) ≤ 0: Sections below the x-axis

    • For P(x) ≥ 0: Sections above the x-axis

Solving Polynomial Inequalities - Sign Table Method

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

  • P(x) ≤ 0

• STEP 2: Find the roots of the polynomial

  • Factorizing to solve P(x) = 0

• STEP 3: Sign table method

  • 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 factorized, it is possible to 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

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