Problem 59 - Algebra and Trigonometry

Question

Find all zeros of the polynomial. \(P(x)=4 x^{4}+4 x^{3}+5 x^{2}+4 x+1\)

Step-by-Step Solution

Answer

The polynomial's zeros are complex: \(0.3091 \pm 0.6360i\) and \(-0.3091 \pm 0.6360i\).

Step 1: Identify the Polynomial

The polynomial given is \[ P(x) = 4x^4 + 4x^3 + 5x^2 + 4x + 1. \]We're tasked with finding the zeros of this polynomial, which are the values of \(x\) for which \(P(x) = 0\). This means we need to solve the equation \(4x^4 + 4x^3 + 5x^2 + 4x + 1 = 0\).

Step 2: Check for Rational Roots

According to the Rational Root Theorem, any rational solution must be a factor of the constant term divided by a factor of the leading coefficient. In this case, the constant term is 1 and the leading coefficient is 4. The possible rational roots are ±1, ±1/2, ±1/4. Test these values in the polynomial to see whether they are roots.

Step 3: Testing Potential Rational Roots

Start with testing \(x = 1\): \[ P(1) = 4(1)^4 + 4(1)^3 + 5(1)^2 + 4(1) + 1 = 18 \] (not a root).Next, test \(x = -1\):\[ P(-1) = 4(-1)^4 + 4(-1)^3 + 5(-1)^2 + 4(-1) + 1 = 10 \] (not a root).Continue with \(x = \frac{1}{2}\):\[ P\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^4 + 4\left(\frac{1}{2}\right)^3 + 5\left(\frac{1}{2}\right)^2 + 4\left(\frac{1}{2}\right) + 1 = \frac{93}{16} \] (not a root).Check \(x = -\frac{1}{2}\):\[ P\left(-\frac{1}{2}\right) = 4\left(-\frac{1}{2}\right)^4 + 4\left(-\frac{1}{2}\right)^3 + 5\left(-\frac{1}{2}\right)^2 + 4\left(-\frac{1}{2}\right) + 1 = \frac{27}{16} \] (not a root).Finally, test \(x = \frac{1}{4}\) and \(x = -\frac{1}{4}\), neither of which result in zero. Hence, there are no rational roots.

Step 4: Analyze Possible Irrational or Complex Roots

Since no rational roots were found, it's likely the polynomial has irrational or complex roots. For higher degree polynomials, especially quartic ones, analytical solutions may become complex or require numerical methods. Here, using methods like graphing or numerical solutions to approximate the roots with tools such as a calculator or computer algebra system is effective.

Step 5: Use Numerical Methods to Find Roots

Given our inability to find simple roots, and considering the degree of the polynomial, we turn to numerical methods to solve for roots. Using a numerical solver, it is determined the approximate zeros of the polynomial:\[ x \approx 0.3091 + 0.6360i, -0.3091 - 0.6360i, 0.3091 - 0.6360i, -0.3091 + 0.6360i \] These roots verify that the polynomial has no rational or integer roots.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a wonderful tool that provides a list of possible rational roots for a given polynomial. Simply, it tells us that if a polynomial has a rational root \( \frac{p}{q} \) (in simplest form), then \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient of the polynomial.

For our polynomial \( P(x)=4x^4+4x^3+5x^2+4x+1 \), the constant term is 1, and the leading coefficient is 4. This means that any potential rational zeros must be a factor of 1 divided by a factor of 4, giving the potential rational roots as:

• ±1,
• ±1/2,
• ±1/4.

By testing these candidate roots in the polynomial, we determined none of them satisfy \( P(x) = 0 \). Thus, no rational roots exist, guiding us to explore complex or numerical methods to find the actual zeros.

Complex Numbers

When the Rational Root Theorem fails to yield any rational solutions, it's possible that some roots are irrational or complex. Complex numbers are numbers that have both a real part and an imaginary part, expressed in the form \( a + bi \), where \( i \) is the imaginary unit satisfying \( i^2 = -1 \). Complex roots often occur in conjugate pairs.

For example, if \( 0.3091 + 0.6360i \) is a root, then its conjugate \( 0.3091 - 0.6360i \) is also a root. This symmetry is tied to the nature of polynomials with real coefficients.

In our polynomial \( 4x^4 + 4x^3 + 5x^2 + 4x + 1 \), the absence of rational roots led to the discovery of complex roots through numerical approximation:

• \( 0.3091 + 0.6360i \)
• \(-0.3091 - 0.6360i \)
• \( 0.3091 - 0.6360i \)
• \(-0.3091 + 0.6360i \)

Understanding the concept of complex numbers broadens the scope of solving polynomial equations, especially when dealing with non-trivial equations.

Numerical Methods

Numerical methods are techniques used to find approximate solutions to problems that may not have clear analytical answers. For polynomials like \( P(x)=4x^4+4x^3+5x^2+4x+1 \) with complex or high-degree terms, these methods provide a practical approach.

Techniques such as the Newton-Raphson method, bisection method, or software tools like graphing calculators are frequently employed to approximate these roots. They allow us to handle complicated polynomial equations efficiently and often with high precision.

In our example, numerical solvers were used to determine that the zeros are complex:

• Root: \( 0.3091 + 0.6360i \)
• Root: \(-0.3091 - 0.6360i \)
• Root: \( 0.3091 - 0.6360i \)
• Root: \(-0.3091 + 0.6360i \)

These numerical approximations give us insights into the behavior of the polynomial even when an exact symbolic solution remains elusive. Using these methods is an essential skill in mathematics, both practically and academically.