Problem 40 - Precalculus: Functions and Graphs

Question

A Chebyshev polynomial The fourth-degree Chebyshev polynomial \(f(x)=8 x^{4}-8 x^{2}+1\) occurs in statistical studies. Find all values of \(x\) such that \(f(x)>0 .\) (Hint: Let \(z=x^{2},\) and use the quadratic formula.)

Step-by-Step Solution

Answer

Values of \( x \) for which \( f(x) > 0 \) are solutions in the intervals: \((-\infty, -\sqrt{\frac{2 + \sqrt{2}}{4}})\), \((-\sqrt{\frac{2 - \sqrt{2}}{4}}, \sqrt{\frac{2 - \sqrt{2}}{4}})\), and \((\sqrt{\frac{2 + \sqrt{2}}{4}}, \infty)\).

Step 1: Substitute and Simplify

Given the polynomial function \( f(x) = 8x^4 - 8x^2 + 1 \), we start by using the substitution \( z = x^2 \). This converts the fourth-degree polynomial into a quadratic function of \( z \): \( f(z) = 8z^2 - 8z + 1 \). We need to find when \( f(z) > 0 \).

Step 2: Solve Quadratic Inequality

Using the quadratic formula, find the roots of \( 8z^2 - 8z + 1 = 0 \). The quadratic formula is given by \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 8 \), \( b = -8 \), and \( c = 1 \). Substituting these values gives \( z = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 8 \cdot 1}}{16} \).

Step 3: Calculate the Discriminant and Roots

Calculate the discriminant: \( b^2 - 4ac = (-8)^2 - 4 \times 8 \times 1 = 64 - 32 = 32 \). Hence, the roots are \( z_1 = \frac{8 + \sqrt{32}}{16} \) and \( z_2 = \frac{8 - \sqrt{32}}{16} \). Simplifying, \( \sqrt{32} = 4\sqrt{2} \), so \( z_1 = \frac{8 + 4\sqrt{2}}{16} = \frac{2 + \sqrt{2}}{4} \) and \( z_2 = \frac{8 - 4\sqrt{2}}{16} = \frac{2 - \sqrt{2}}{4} \).

Step 4: Analyze the Inequality

The inequality \( 8z^2 - 8z + 1 > 0 \) implies \( z < z_2 \) or \( z > z_1 \) because the parabola opens upwards (coefficient of \( z^2 \) is positive). Converting intervals for \( z \) back to \( x^2 \): \( x^2 < \frac{2 - \sqrt{2}}{4} \) or \( x^2 > \frac{2 + \sqrt{2}}{4} \).

Step 5: Find the Values of x

For \( x^2 < \frac{2 - \sqrt{2}}{4} \), the values of \( x \) are \( -\sqrt{\frac{2 - \sqrt{2}}{4}} < x < \sqrt{\frac{2 - \sqrt{2}}{4}} \). For \( x^2 > \frac{2 + \sqrt{2}}{4} \), the solution is \( x < -\sqrt{\frac{2 + \sqrt{2}}{4}} \) or \( x > \sqrt{\frac{2 + \sqrt{2}}{4}} \).

Step 6: Conclusion

The solution set for \( x \) such that \( f(x) > 0 \) is \( x \in (-\infty, -\sqrt{\frac{2 + \sqrt{2}}{4}}) \cup (-\sqrt{\frac{2 - \sqrt{2}}{4}}, \sqrt{\frac{2 - \sqrt{2}}{4}}) \cup (\sqrt{\frac{2 + \sqrt{2}}{4}}, \infty) \).

Key Concepts

Quadratic Inequality

A quadratic inequality can be thought of as a comparison between a quadratic expression and zero. In simple terms, we want to determine when the quadratic is greater than zero, less than zero, or possibly equal to zero. For the Chebyshev polynomial turned quadratic function

• Step 1: Identify the inequality format, which in this case is expressed as \(8z^2 - 8z + 1 > 0\).
• Step 2: Solve for when this inequality holds true. Generally, this involves finding the roots of the corresponding equation and analyzing the intervals created by these roots.

We seek the intervals where the quadratic function is positive. Translating this to the context of a parabola, this condition holds in the ranges where the parabola opens upwards and outside the interval of the roots.

Quadratic Formula

The quadratic formula is an essential tool used to solve equations of the form \(ax^2 + bx + c = 0\). It enables the calculation of the roots of any quadratic equation using the formula: \[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This provides the solutions, or x-intercepts, for any quadratic equation. For our Chebyshev polynomial exercise translated to

• \(a = 8\), \(b = -8\), and \(c = 1\).
• Using these values in the quadratic formula allows the determination of the precise roots, which become vital in understanding the sign changes of the function.

The presence of the roots helps divide the number line into sections where the quadratic expression may be positive or negative.

Polynomial Inequalities

Polynomial inequalities are a bit like a puzzle. They involve understanding on which intervals a polynomial (which can be a quadratic, cubic, or even higher degree) is greater or less than zero. For quadratic inequalities

• Roots are pivotal because they act as boundaries for interval checking.
• You analyze the sign of the polynomial in segments defined by these roots.

In this exercise, the inequality \(8z^2 - 8z + 1 > 0\) was solved by finding the roots and testing the intervals they define. This analysis showed in which segments the polynomial is positive by considering the mathematical properties of parabolas. The parabola of this expression, as derived from its quadratic form, opens upwards, as shown by a positive leading coefficient.

Discriminant Calculation

When using the quadratic formula, a discriminant calculation is crucial as it informs us about the nature of the roots. The formula for the discriminant is \(b^2 - 4ac\). In essence, this value can tell you:

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, meaning the roots are equal.
• If \(b^2 - 4ac < 0\), the solutions are complex, and therefore, not real.

In our exercise, the discriminant was \(32\), which is positive, indicating two distinct real roots. These roots help form critical interval endpoints that guide us in analyzing where the polynomial is greater than zero.