Problem 14 - Algebra and Trigonometry

Question

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{3}-8\)

Step-by-Step Solution

Answer

The zeros of \(P(x)\) are \(x = 2, -1 \pm i\sqrt{3}\). Complete factorization: \((x-2)(x+1-i\sqrt{3})(x+1+i\sqrt{3})\).

Step 1: Recognize the Form

The polynomial given is a cubic polynomial, specifically a difference of cubes: \(x^3 - 8\). We can express \(8\) as \(2^3\), which allows us to write the polynomial as \(x^3 - 2^3\).

Step 2: Apply the Difference of Cubes Formula

The formula for the difference of cubes is: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = x\) and \(b = 2\). Applying this formula, we get: \(P(x) = (x-2)(x^2 + 2x + 4)\).

Step 3: Solve for Real Zeros

The real zero can be found from the \(x-2\) factor. Setting it to zero gives \(x - 2 = 0\), which results in \(x = 2\).

Step 4: Solve the Quadratic for Complex Zeros

Now, solve the quadratic equation \(x^2 + 2x + 4 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 2\), and \(c = 4\). This gives \(x = \frac{-2 \pm \sqrt{4 - 16}}{2}\), which simplifies to \(x = \frac{-2 \pm \sqrt{-12}}{2}\).

Step 5: Simplify the Complex Solutions

The term \(\sqrt{-12}\) can be written as \(\sqrt{12}i = 2\sqrt{3}i\). Thus, the solutions are \(x = -1 \pm i\sqrt{3}\).

Step 6: Write the Complete Factorization

The complete factorization of \(P(x)\) is:\[P(x) = (x - 2)(x - (-1 + i\sqrt{3}))(x - (-1 - i\sqrt{3}))\]

Key Concepts

Difference of Cubes

Understanding the difference of cubes is crucial when dealing with cubic polynomials like the one given, \(P(x) = x^3 - 8\). The difference of cubes follows a specific pattern or formula:

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

Identifying \(a\) and \(b\) is the first step. In this case \(a = x\) and \(b = 2\), since the expression \(8\) can be rewritten as \(2^3\).
Applying the formula, the polynomial \(x^3 - 8\) can be decomposed into \((x - 2)(x^2 + 2x + 4)\). Here, the term \(x^2 + 2x + 4\) requires further solving to determine its zeros.
This decomposition allows us to simplify the polynomial and make it easier to solve for both real and complex zeros.

Complex Zeros

Complex zeros occur when the solutions of a polynomial equation are not real numbers, typically when you have a negative inside a square root during calculations. When dealing with polynomials after factorization, complex zeros often emerge from the quadratic portion that cannot be simplified to real solutions.

• For \(P(x) = x^3 - 8\), the factorization led to \((x - 2)(x^2 + 2x + 4)\).

The real solution is straightforward from the \((x-2)\) factor. However, the quadratic \(x^2 + 2x + 4 = 0\) requires using the quadratic formula to find the other zeros.
Applying the formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = 1\), \(b = 2\), and \(c = 4\). The discriminant \(b^2 - 4ac\) results in a negative number, \(4 - 16 = -12\).
This indicates complex solutions:

• \(x = -1 + i\sqrt{3}\)
• \(x = -1 - i\sqrt{3}\)

Each solution has a real part and an imaginary part, expressed as \(i\) times the square root of a positive number.

Quadratic Formula

The quadratic formula is a useful tool for finding the zeros of any quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by:

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

In solving the polynomial \(P(x) = x^3 - 8\), we apply this formula to the quadratic piece \(x^2 + 2x + 4\) after factorization. The constants are identified as \(a=1\), \(b=2\), and \(c=4\).
The challenge arises when the discriminant \(b^2 - 4ac\) is negative, as it results in complex numbers rather than real numbers. This indicates that the parabola described by the quadratic never crosses the x-axis.
With \(b^2 - 4ac = 4 - 16 = -12\), you compute the imaginary part using the square root of the negative value. This is simplified to form the complex zeros: \(x = -1 \pm i\sqrt{3}\).
Understanding this formula not only helps solve quadratic equations but also confirms the existence of complex solutions when the discriminant is negative.