Problem 5 - Precalculus Mathematics for Calculus

Question

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{4}+4 x^{2}$$

Step-by-Step Solution

Answer

Zeros: \(x = 0, \pm 2i\). Factorization: \(x^2(x^2 + 4)\).

Step 1: Set the Polynomial Equal to Zero

First, set the polynomial function equal to zero to find the solutions for zero: \(P(x) = x^{4} + 4x^{2} = 0 \)

Step 2: Factor the Polynomial

Notice that the polynomial expression \(x^{4} + 4x^{2}\) can be factored. Start by factoring out the greatest common factor (GCF), which is \(x^2\):\(x^2(x^2 + 4) = 0\). This gives us two factors: \(x^2\) and \(x^2 + 4\).

Step 3: Solve for Zeros of Each Factor

Now, solve each factor for zeros individually:1. \(x^2 = 0\) results in \(x = 0\).2. \(x^2 + 4 = 0\) leads to \(x^2 = -4\). This can further be solved as \(x = \pm 2i\).

Step 4: List All Zeros

From the calculations in the previous steps, the zeros of the polynomial are:\(x = 0\) and \(x = \pm 2i\).

Step 5: Write the Complete Factorization

Using the zeros found in the previous steps, the complete factorization of \(P(x)\) is:\(P(x) = x^2(x - 2i)(x + 2i)\).Factoring out the imaginary parts, we see: \(P(x) = x^2(x^2 + 4)\).

Key Concepts

real zeros

Real zeros of a polynomial are the solutions where the graph of the polynomial touches or crosses the x-axis. These zeros can be found where the polynomial equation equals zero. For the polynomial \( P(x) = x^4 + 4x^2 \), the real zeros were identified by first factoring the equation. The factorization process showed us that \( x^2 \) is a factor, which itself equals zero at \( x = 0 \).

• Setting \( x^2 = 0 \), we find \( x = 0 \) as the real zero.

Although in this exercise, we find only \( x = 0 \) as a real zero, many polynomials have more than one real zero. To find these, one would typically factor the polynomial completely and identify all roots that are real numbers. Understanding real zeros is crucial as they give us direct intersections with the x-axis on the graph, representing the input values that make the polynomial equal to zero.

complex zeros

Complex zeros are solutions to polynomial equations that include imaginary numbers, usually found when real solutions are not possible. These zeros occur in conjugate pairs and are particularly useful for understanding the full behavior of polynomial functions.
For \( P(x) = x^4 + 4x^2 \), after factoring, we find the expression \( x^2 + 4 \) which leads to the equation \( x^2 = -4 \). Solving this gives complex zeros because negative numbers under a square root lead to imaginary numbers. The solutions are:

• \( x = 2i \)
• \( x = -2i \)

These zeros are complex because they involve \( i \), the imaginary unit where \( i^2 = -1 \). Complex zeros often don't correspond to actual points on the x-axis, but they still hold valuable information. They tell us about the symmetry and the nature of the polynomial, expanding our view beyond just the real-number system.

factoring polynomials

Factoring polynomials is a method used to express the polynomial as a product of its simpler factors. This process is integral in algebra as it reveals the zeros of the polynomial and helps in simplifying expressions. The factorization of \( P(x) = x^4 + 4x^2 \) begins with identifying a common factor: \( x^2 \).

• The polynomial becomes \( x^2(x^2 + 4) \).

This breaks \( P(x) \) into simpler parts, which are easier to solve. Factoring is akin to reverse-engineering the multiplication that created the polynomial. In our exercise, this factorization is helpful not only for finding zeros but also for simplifying the expression into a form that highlights its core characteristics.
Understanding how to factor correctly ensures that we fully break down a polynomial, make it easier to handle, and allow a full understanding of its structure and behavior.