Problem 7 - Algebra and Trigonometry

Question

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{4}-16 $$

Step-by-Step Solution

Answer

Zeros: \( x = 2, -2, 2i, -2i \). Factored: \( (x-2)(x+2)(x-2i)(x+2i) \).

Step 1: Recognize as Difference of Squares

Notice that the polynomial \( P(x) = x^4 - 16 \) is a difference of squares, since it can be expressed in the form \( a^2 - b^2 \). Here, \( a = x^2 \) and \( b = 4 \).

Step 2: Apply Difference of Squares Formula

Use the formula for the difference of squares: \( a^2 - b^2 = (a-b)(a+b) \). Substitute to get \( (x^2 - 4)(x^2 + 4) \).

Step 3: Further Factorization

Factor \( x^2 - 4 \) again as it is also a difference of squares, \( x^2 - 4 = (x - 2)(x + 2) \). The polynomial now becomes \( (x - 2)(x + 2)(x^2 + 4) \).

Step 4: Find Zeros of Real Factors

Set each real factor equal to zero. For \( x - 2 = 0 \), \( x = 2 \). For \( x + 2 = 0 \), \( x = -2 \). These are the real zeros.

Step 5: Solve for Complex Zeros

Set the remaining quadratic factor equal to zero: \( x^2 + 4 = 0 \). Solve for \( x \) to find: \( x^2 = -4 \). Taking the square root gives \( x = \pm 2i \). These are the complex zeros.

Step 6: Write the Complete Factorization

The complete factorization of \( P(x) \) is \( (x - 2)(x + 2)(x - 2i)(x + 2i) \), including both real and complex factors.

Key Concepts

Understanding Complex Numbers

Complex numbers extend the idea of the one-dimensional number line. They allow us to solve equations that have no real roots, like all those ending in negative square roots. A complex number consists of two parts:

• The *real part*, which can be any number along the usual number line (like 3 or -1).
• The *imaginary part*, which is based on the imaginary unit "i"—defined as the square root of -1.

Together, they form a complex number such as 3 + 2i. In our original problem, when solving the equation \( x^2 = -4 \), we found that the square roots of -4 are \( \pm 2i \). Without complex numbers, we'd be at a loss for solutions, since no real number squared gives us a negative result.
Using complex numbers, we can extend factorization and see that every polynomial can be completely factored in the complex number system. This concept is fundamental and allows us to find solutions even when there are no real ones.

Difference of Squares in Polynomials

The difference of squares is a vital concept in the world of algebra and polynomial factorization. It simplifies expressions and comes in handy when solving polynomial equations. The difference of squares takes the form \( a^2 - b^2 \) and is expressed mathematically as \((a - b)(a + b)\).
Consider our polynomial from the exercise: \( P(x) = x^4 - 16 \). Recognizing it as a difference of squares lets us rewrite it as \( (x^2)^2 - 4^2 \). Here, \( a = x^2 \) and \( b = 4 \).
Applying the formula, we can break it down further to \( (x^2 - 4)(x^2 + 4) \). The process continues as we notice \( x^2 - 4 \) is itself a difference of squares, which can be further factored into \((x - 2)(x + 2)\).
This method is a powerful tool for simplifying polynomials, as it enables us to reduce even a high-degree polynomial like ours, \( x^4 - 16 \), into simpler, more manageable pieces.

Zeros of Polynomials Explained

Zeros of polynomials are the values of the variable that make the polynomial equal to zero. These zeros can be real or complex.
Let's break down how to find these zeros using our polynomial \( P(x) = x^4 - 16 \). We factored it previously into \((x - 2)(x + 2)(x^2 + 4)\). The next step is setting each factor equal to zero:

• For \(x - 2 = 0\), we find a zero at \(x = 2\).
• For \(x + 2 = 0\), we find another zero at \(x = -2\).
• For the quadratic factor \(x^2 + 4 = 0\), solving for \(x\) leads us to the complex zeros \(x = \pm 2i\).

Finding the zeros of a polynomial is crucial since it helps us understand the polynomial's behavior. Each zero corresponds to an x-intercept if real, or lies in the complex plane if imaginary. Knowing the zeros helps us fully factor the polynomial and predict where the function will cross the x-axis. This is key in many mathematical and real-world applications.