Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 15: More Inequalities

Lesson 15.2: Beyond Quadratics

In this Grade 4 AoPS Introduction to Algebra lesson from Chapter 15, students learn how to solve polynomial and rational inequalities with three or more factors, such as cubic expressions and rational expressions with factors in the denominator. Using sign charts and number line analysis, they determine where products and quotients are positive or negative by tracking the sign of each individual factor across different intervals. The lesson is part of the AMC 8 and 10 preparation curriculum and builds directly on earlier quadratic inequality techniques.

Section 1

Review: Quadratic Inequalities with Sign Analysis

Property

To solve a quadratic inequality using sign analysis:

  1. Write the inequality in standard form: One side is zero, and the other has the form ax2+bx+cax^2 + bx + c.
  2. Find the zeros of ax2+bx+cax^2 + bx + c by setting it equal to zero and solving for xx.
  3. Use the zeros to divide the number line into intervals.
  4. Test the sign of ax2+bx+cax^2 + bx + c in each interval to determine where the inequality is satisfied.

Examples

Section 2

Factoring Higher-Degree Polynomials

Property

To solve polynomial inequalities of degree 3 or higher, factor the polynomial completely into linear and irreducible quadratic factors: P(x)=a(x−r1)(x−r2)...(x2+bx+c)P(x) = a(x - r_1)(x - r_2)...(x^2 + bx + c) where each factor contributes to the sign analysis.

Examples

Section 3

Polynomial Inequalities with Multiple Factors

Property

For polynomial inequalities with multiple factors, solve by finding zeros of each factor, creating intervals, and using sign analysis. The inequality (x−a)(x−b)(x−c)>0(x - a)(x - b)(x - c) > 0 is satisfied when an odd number of factors are negative, while (x−a)(x−b)(x−c)<0(x - a)(x - b)(x - c) < 0 is satisfied when an even number of factors are negative.

Examples

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Chapter 15: More Inequalities

  1. Lesson 1Current

    Lesson 15.2: Beyond Quadratics

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  2. Lesson 2

    Lesson 15.3: The Trivial Inequality

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  3. Lesson 3

    Lesson 15.4: Quadratic Optimization

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Section 1

Review: Quadratic Inequalities with Sign Analysis

Property

To solve a quadratic inequality using sign analysis:

  1. Write the inequality in standard form: One side is zero, and the other has the form ax2+bx+cax^2 + bx + c.
  2. Find the zeros of ax2+bx+cax^2 + bx + c by setting it equal to zero and solving for xx.
  3. Use the zeros to divide the number line into intervals.
  4. Test the sign of ax2+bx+cax^2 + bx + c in each interval to determine where the inequality is satisfied.

Examples

Section 2

Factoring Higher-Degree Polynomials

Property

To solve polynomial inequalities of degree 3 or higher, factor the polynomial completely into linear and irreducible quadratic factors: P(x)=a(x−r1)(x−r2)...(x2+bx+c)P(x) = a(x - r_1)(x - r_2)...(x^2 + bx + c) where each factor contributes to the sign analysis.

Examples

Section 3

Polynomial Inequalities with Multiple Factors

Property

For polynomial inequalities with multiple factors, solve by finding zeros of each factor, creating intervals, and using sign analysis. The inequality (x−a)(x−b)(x−c)>0(x - a)(x - b)(x - c) > 0 is satisfied when an odd number of factors are negative, while (x−a)(x−b)(x−c)<0(x - a)(x - b)(x - c) < 0 is satisfied when an even number of factors are negative.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 15: More Inequalities

  1. Lesson 1Current

    Lesson 15.2: Beyond Quadratics

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  2. Lesson 2

    Lesson 15.3: The Trivial Inequality

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  3. Lesson 3

    Lesson 15.4: Quadratic Optimization

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