Problem 6 - Precalculus Mathematics for Calculus

Question

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{5}+9 x^{3}$$

Step-by-Step Solution

Answer

Zeros: 0 (triple), 3i, -3i; Factored form: \(x^3(x - 3i)(x + 3i)\).

Step 1: Identify Common Factor

First, observe that the terms of the polynomial \(x^5 + 9x^3\) share a common factor. Factor out \(x^3\) from the polynomial: \[P(x) = x^3(x^2 + 9)\]

Step 2: Solve for Real Zeros

To find the zeros of \(P(x)\), set each factor equal to zero. Start with solving \(x^3 = 0\). This gives us the real zeros:\[x^3 = 0 \Rightarrow x = 0\]

Step 3: Solve for Complex Zeros

Next, solve for the zeros of the quadratic factor: \(x^2 + 9 = 0\). Subtract 9 from both sides:\[x^2 = -9\]Take the square root of both sides to find the complex zeros:\[x = \pm 3i\]

Step 4: List All Zeros

Having derived the solutions from Steps 2 and 3, list all zeros of the polynomial:\(x = 0\) (repeated three times), \(x = 3i\), and \(x = -3i\).

Step 5: Factor the Polynomial Completely

Use the zeros to write the polynomial as a product of its factors:\[P(x) = x^3(x - 3i)(x + 3i)\]This expresses the polynomial completely factored over the complex numbers.

Key Concepts

Real and Complex Zeros

When dealing with polynomials, finding zeros is a crucial step. Zeros of a polynomial, also called roots, are the values for which the polynomial evaluates to zero. In this exercise, we dealt with the polynomial \( P(x) = x^5 + 9x^3 \). Zeros can be either real or complex, and understanding these is fundamental in algebra.

• Real Zeros: Real zeros are the solutions to the polynomial equation that are real numbers. They graphically represent the points where the polynomial curve intersects the x-axis.
• Complex Zeros: These are zeros that include imaginary numbers, involving the imaginary unit \( i \) where \( i^2 = -1 \). They do not appear on the real plane in a graph but are vital in expressing polynomials in their simplest form.

In our problem, the real zero is found by solving \( x^3 = 0 \), resulting in \( x = 0 \), whereas the complex zeros are found by solving \( x^2 + 9 = 0 \), giving us \( x = \pm 3i \).

Polynomial Roots

Understanding polynomial roots is the key to solving polynomial equations. In essence, roots are the x-values that make the whole polynomial equal zero. For any given polynomial of degree \( n \), there are at most \( n \) roots. This includes both real and complex zeros.

• Multiplicity of Roots: A root that occurs more than once in the solution set has a multiplicity greater than one. In our problem, \( x = 0 \) is a root of multiplicity three, meaning it appears three times.
• Graphical Representation: Roots influence how the graph behaves. For example, a root with odd multiplicity will cross the x-axis, whereas a root with even multiplicity will touch the x-axis and bounce back.

In the given polynomial \( P(x) = x^5 + 9x^3 \), the root \( x = 0 \) touches and crosses the x-axis multiple times, while the complex roots \( x = 3i \) and \( x = -3i \) do not affect the real plot but are critical for factorization.

Factorization Techniques

Polynomials can be factored into simpler components that are easier to work with. Factorization converts a polynomial into a product of its factors, which could either be numbers, variables, or both.

• Common Factor Extraction: Always check for common factors in the terms of a polynomial. In \( P(x) = x^5 + 9x^3 \), \( x^3 \) is the common factor extracted first, simplifying the expression to \( x^3(x^2 + 9) \).
• Solving Quadratics: Once simplified, solving remaining simple quadratics or binomials can give further zeros. For example, \( x^2 + 9 \) was solved next to find complex zeros.
• Complete Factorization: By using all derived factors and zeros, the polynomial can be completely factored. This enables us to write \( P(x) \) as \( x^3(x - 3i)(x + 3i) \), fully expressing the polynomial.

Factorization is essential not only for finding zeros but also for simplifying polynomial expressions and solving polynomial equations efficiently.