Problem 55 - A Graphical Approach to Precalculus with Limits

Question

We examine how the three complex cube roots of \(-8\) can be found in two different ways. All complex roots of the equation \(x^{3}+8=0\) are cube roots of \(-8 .\) Factor \(x^{3}+8\) as the sum of two cubes.

Step-by-Step Solution

Answer

The cube roots of \(-8\) are \(-2, 1 + i\sqrt{3},\) and \(1 - i\sqrt{3}.\)

Step 1: Recognize the form of the expression

The given equation is \(x^3 + 8 = 0\). This can be rewritten as \(x^3 + 2^3\). Recognizing this as a sum of cubes is crucial for factoring.

Step 2: Use the sum of cubes formula

The formula for factoring a sum of cubes is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 2\).

Step 3: Apply the formula to factor

Substitute \(a = x\) and \(b = 2\) into the sum of cubes formula: \(x^3 + 8 = (x + 2)(x^2 - 2x + 4)\). This gives us the factors of the original cubic equation.

Step 4: Solve the linear factor

Set the linear factor equal to zero: \(x + 2 = 0\). Solving for \(x\) gives \(x = -2\), which is one root of the equation.

Step 5: Solve the quadratic factor

Set the quadratic factor equal to zero: \(x^2 - 2x + 4 = 0\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = -2\), and \(c = 4\).

Step 6: Calculate the discriminant

Calculate the discriminant of the quadratic: \((-2)^2 - 4(1)(4) = 4 - 16 = -12\). The negative discriminant indicates complex roots.

Step 7: Use the quadratic formula

Plug the values into the quadratic formula: \(x = \frac{-(-2) \pm \sqrt{-12}}{2(1)} = \frac{2 \pm \sqrt{12}i}{2}\). Simplifying further gives \(x = 1 \pm i\sqrt{3}\).

Step 8: List all roots

The three roots of \(x^3 + 8 = 0\) are \(x = -2\), \(x = 1 + i\sqrt{3}\), and \(x = 1 - i\sqrt{3}\). These are the cube roots of \(-8\).

Key Concepts

Sum of Cubes

The sum of cubes is a process of factoring an expression that is structured in the form of two perfect cubes being added together, written as \(a^3 + b^3\). This form allows us to use a specific formula to break the expression into a product of two simpler expressions. The key formula is:

• \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

Recognizing this structure is critical because it simplifies complex expressions, making them easier to solve. For our example, \(x^3 + 8\), which can be rewritten as \(x^3 + 2^3\), follows the sum of cubes structure. By identifying \(a = x\) and \(b = 2\), we can apply this formula to factor it into \((x + 2)(x^2 - 2x + 4)\). Understanding this algebraic strategy helps in solving polynomial equations easily.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The solution involves plugging the coefficients of the quadratic equation into the formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a\), \(b\), and \(c\) are coefficients from the equation. In solving the quadratic factor \(x^2 - 2x + 4 = 0\) obtained from our sum of cubes, we identify \(a = 1\), \(b = -2\), and \(c = 4\). Before applying the formula, it's vital to compute the discriminant \(b^2 - 4ac\). In this problem, the discriminant results in a negative value, leading us into the world of complex roots. This illustrates the versatility of the quadratic formula in handling both real and complex numbers.

Complex Roots

Complex roots arise when solving equations yields a negative discriminant, indicating numbers that cannot be plotted on the traditional number line. Essentially, a negative discriminant from the quadratic formula, like our previous example where \(b^2 - 4ac = -12\), signals complex roots. Complex numbers usually take the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, fulfilling \(i^2 = -1\).
For the quadratic \(x^2 - 2x + 4 = 0\), solving it using the quadratic formula yields the complex roots \(x = 1 \pm i\sqrt{3}\). These roots are not only mathematically interesting but also necessary for fully describing the behavior of functions related to the equation \(x^3 + 8 = 0\). Understanding complex roots expands our view of traditional number sets, enabling powerful solutions across various mathematical problems.