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Chapter 3: Problem 26

Solve the polynomial inequality. $$x^{3}<4 x^{2}-4 x$$

Short Answer

Expert verified
The solutions to the polynomial inequality are \( x<0 \), \( 14 \).

Step by step solution

01

Rewrite the inequality

Rearrange the inequality to form it equal to zero, obtaining \( x^3 - 4x^2 + 4x < 0 \).
02

Factor the polynomial

Factor the polynomial in the equation. It simplifies to \( x(x-1)(x-4) < 0 \).
03

Find the roots

Set \( x(x-1)(x-4) = 0 \) to find the roots. The roots are 0, 1, and 4.
04

Test the intervals

Test the intervals of the roots in the factored inequality. For \( x<0 \), for example choose \( x=-1 \), the inequality becomes true (negative). For \( 04 \) will yield a true inequality (negative).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is the process of breaking down a complex polynomial into simpler pieces, called factors, that when multiplied give back the original polynomial. This is akin to breaking up a sentence into words 鈥� it makes handling complex expressions much simpler.
  • The first step is to set your polynomials to equal zero (as seen in the exercise) 鈥� this is crucial for identifying key values or roots later.

  • Next, extract common factors. In our example, notice that each term contains an 'x', so you factor it out first.

  • Then, continue factoring the remaining polynomial until no further factoring is possible. Using techniques like grouping or simple trinomial factoring helps here.
Once the polynomial from the exercise is factored, you derived: \( x(x-1)(x-4) < 0 \). This method dramatically simplifies the investigation of the behavior of the polynomial.
Roots of Polynomial Equations
Roots of polynomial equations are the values that make the entire polynomial equal to zero. Finding these roots is essential as they divide the number line into intervals that can be tested.
  • To find the roots, set the factored polynomial equal to zero: \( x(x-1)(x-4) = 0 \).

  • Each factor must be zero for the entire expression to be zero:
    • \( x = 0 \)
    • \( x - 1 = 0 \) leads to \( x = 1 \)
    • \( x - 4 = 0 \) leads to \( x = 4 \)

  • These roots are not just solutions, but also key points to divide the polynomial鈥檚 behavior across the number line.
The roots in this specific case are 0, 1, and 4. Each root offers a boundary where the inequality changes its truth value.
Testing Intervals of Inequalities
Testing intervals of inequalities is the step to determine where a certain inequality holds true or false along the number line. After finding the roots, the number line is split into distinct regions which are tested individually.
  • Identify intervals between roots: In our example, intervals are \( x<0 \), \( 04 \).

  • Test points from within each interval in the inequality \( x(x-1)(x-4) < 0 \):
    • For example, if tested \( x = -1 \), within \( x < 0 \), the inequality is true.
    • If tested \( x = 0.5 \), between \( 0 < x < 1 \), the inequality is false.

  • Determine which intervals are true: You find that \( x<0 \), \( 14 \) satisfy the inequality, as seen in the solution steps.
This testing process confirms in which sections of the number line the original inequality is true. Therefore, you can sketch out the solutions or apply them to contextual problems.

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