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Magazine Chapter 2: Problem 31
Solving a Polynomial Inequality In Exercises \(13-34,\) solve the inequality. Then graph the solution set.$$x^{3}-4 x \geq 0$$
Short Answer
Expert verified
The solution set of the inequality \(x^{3}-4 x \geq 0\) is \(-\infty < x \leq -2\), \( 2 \leq x < \infty\).
Step by step solution
01
Simplify the Expression and Factorize
In order to simplify the expression, factorize \(x^{3}-4 x\).\nSo, \(x^{3}-4 x=x(x^{2}-4)\).\nThis can be further factorized by using difference of squares (a-b)(a+b),\nSo, we get \(x(x-2)(x+2)\).\n
02
Determine the Critical Points
The roots of the equation are the x-intercepts. These roots are found by setting the expressions equal to zero.\nSo, \(x(x-2)(x+2)=0\)\nThis gives us the roots \(x=0\), \(x=2\) and \(x=-2\). These are the critical points.
03
Determine the Intervals
The critical points divide the real number line into four intervals: \n (-鈭�, -2), (-2, 0), (0, 2), and (2, 鈭�).
04
Test each Intervals
Test each of these intervals in the original inequality \(x^{3}-4 x \geq 0\) by selecting a test point in each interval. If the inequality is true, then the entire interval is part of the solution set.\nFor example, choose -3 for the interval (-Inf, -2) in the inequality, the result is positive, so the interval (-Inf, -2) is part of the solution.\nNext, choose -1 for the interval (-2, 0), the result is negative, so the interval (-2, 0) is not part of the solution.\nChoose 1 for the interval (0, 2), the result is negative, so the interval (0, 2) is not part of the solution.\nLastly, choose 3 for the interval (2, Inf), the result is positive, so the interval (2, Inf) is part of the solution.
05
Graph the Solution Set
Draw the number line and mark the critical points on it. For the solution intervals (-鈭�, -2] and [2, 鈭�), color the number line to the left of -2 and to the right of 2. Note that the critical points are included in the solution since the inequality symbol is 鈥樷墺鈥�.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factorization
Factorization is an essential step in solving polynomial inequalities, like the one presented in the exercise: \(x^3 - 4x \geq 0\). Factorization helps simplify equations by breaking them into their component parts. In this case, the polynomial can be factorized by first taking out the common factor of \(x\).
For the expression \(x^3 - 4x\), we identify \(x\) as a common factor.
For the expression \(x^3 - 4x\), we identify \(x\) as a common factor.
- The initial factorization step yields \(x(x^2 - 4)\).
- Further factorizing \(x^2 - 4\) involves recognizing it as a difference of squares.
- This gives \(x(x-2)(x+2)\), indicating that our original polynomial can be rewritten as the product of three simpler polynomials.
Critical Points
Critical points are the values of \(x\) that make each factor in a factorized polynomial equal to zero. This is necessary to understand which parts of the inequality are zero or change sign. These points are where the polynomial might shift from positive to negative (or vice versa), crucial for solving inequalities.
Once the polynomial is factorized as \(x(x-2)(x+2)\), identify when each factor is zero:
Once the polynomial is factorized as \(x(x-2)(x+2)\), identify when each factor is zero:
- \(x = 0\) makes the factor \(x\) zero.
- \(x = 2\) makes the factor \(x-2\) zero.
- \(x = -2\) makes the factor \(x+2\) zero.
Interval Testing
Interval testing involves evaluating the polynomial in the regions between critical points to determine where the inequality holds true. Each critical point splits the number line into distinct intervals:
For example:
- \((-\infty, -2)\)
- \((-2, 0)\)
- \((0, 2)\)
- \((2, \infty)\)
For example:
- In \((-\infty, -2)\), take \(x = -3\). Substituting gives a positive result, meaning this interval satisfies the inequality.
- For \((-2, 0)\), test \(x = -1\). The result is negative, indicating this interval is not part of the solution.
- Use \(x = 1\) for \((0, 2)\), yielding a negative result. Thus, this interval does not satisfy the inequality.
- In \((2, \infty)\), choose \(x = 3\). The positive result indicates this interval fulfills the inequality.
Number Line Graphing
Graphing the solution set on a number line visually represents the intervals where the inequality holds true. This is an important step to clearly communicate the results.
Begin by marking the critical points \(-2, 0,\) and \(2\) on the number line. These points are the boundaries between intervals:
Begin by marking the critical points \(-2, 0,\) and \(2\) on the number line. These points are the boundaries between intervals:
- Use closed circles at \(-2\) and \(2\) because they're included in the solution set due to the 'greater than or equal' condition.
- The intervals \((-\infty, -2]\) and \([2, \infty)\) are part of the solution, so these are shaded on the number line.
- Open circles or gaps are placed at \(0\) since it is not part of the solution intervals.
