Problem 50 - College Algebra

Question

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ 3 x^{4}-11 x^{3}-3 x^{2}-6 x+8-0 $$

Step-by-Step Solution

Answer

The zeros of the given polynomial function are \(x = 1, -1/3, -2, 2i\).

Step 1: Applying the Rational Zero theorem

Use the Rational Zero theorem which states that any potential rational root, p/q, should be a factor of the constant term divided by a factor of the leading coefficient. Here, the constant term is 8 and the leading coefficient is 3. So, the potential roots could be ±1, ±2, ±4, ±8 and ±1/3, ±2/3, ±4/3, ±8/3.

Step 2: Applying Descartes' Rule of Signs

Descartes' Rule of Signs helps us estimate the number of positive and negative real roots in a polynomial. This can limit the number of potential roots we need to check. It works by counting the number of sign changes in the polynomial. The polynomial here is \(3x^4 - 11x^3 -3x^2 -6x + 8 = 0\), and as we can see, the signs alternate four times. So, the number of positive real roots is either 4 or 4 - (even number), which leaves us with 4 or 2. If we replace \(x\) with \(-x\), the polynomial becomes \(3x^4 + 11x^3 - 3x^2 + 6x + 8\), which has two sign changes, thus indicating 2 or 0 negative real roots.

Step 3: Testing potential roots and obtaining real roots

Now that we have an idea of the number of possible real roots through Descartes' Rule of Signs, we can test the potential roots obtained from the Rational Zero theorem. The roots of the polynomial are found to be \(x = 1, -1/3, -2\) after testing potential roots by substituting in the equation.

Step 4: Checking for remaining roots

As one of the roots found is not rational, the cubic formula or a graphing utility can be used to confirm the remaining root(s). If there are any left, they will be complex roots. In this case, after testing, we find that the remaining root is a complex root, \(x = 2i\).

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a powerful tool in polynomial equations, especially for identifying potential rational roots of a polynomial function. This theorem asserts that any potential rational solution or zero of a polynomial, can be expressed as \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

• In our given polynomial \( 3x^4 - 11x^3 - 3x^2 - 6x + 8 = 0 \), the constant term is 8 and the leading coefficient is 3.
• The factors of 8 are ±1, ±2, ±4, ±8, and the factors of 3 are ±1, ±3.
• This leads to potential rational roots of ±1, ±2, ±4, ±8 and ±1/3, ±2/3, ±4/3, ±8/3.

By substituting these potential zeros into the polynomial, we can quickly identify which, if any, are actual zeros. This systematic approach significantly narrows down the search for rational roots in polynomial equations.

Descartes' Rule of Signs

Descartes' Rule of Signs is an efficient method to estimate the number of positive and negative real roots a polynomial has. It relies on analyzing the sign changes in the coefficients of a polynomial when arranged in descending order of power.

• For positive roots, observe the number of sign changes in the original polynomial expression.
• In our equation \(3x^4 - 11x^3 - 3x^2 - 6x + 8 = 0\), there's a total of four sign changes, indicating between four and zero positive real roots.
• Evaluate for negative roots by substituting \( x \) with \( -x \), and count the sign changes again.
• After substituting, \(3x^4 + 11x^3 - 3x^2 + 6x + 8\) has two sign changes, predicting two or zero negative real roots.

This rule helps prioritize which potential roots to verify and enhances our understanding of the polynomial's behavior, limiting unnecessary calculations.

Real Roots

Real roots of a polynomial are the solutions that fall on the real number line, where the polynomial equals zero. Identifying real roots allows you to understand intersections of the polynomial graph with the x-axis. Through techniques like the Rational Zero Theorem and Descartes' Rule of Signs, finding these roots becomes more manageable.

• From our polynomial, known real roots were found to be \( x = 1, -1/3, -2 \) by substituting potential zeros into the equation and verifying equivalence to zero.
• Verifying involves calculating within the polynomial function — ensuring the output is zero for actual roots.

Real roots represent tangible solutions, making sense in many real-world scenarios, like when calculating physical measurements or object intersections.

Complex Roots

Complex roots occur when a polynomial equation does not intersect the x-axis on a real number line but exists in the complex plane. They are composed of a real part and an imaginary part, with the imaginary part involving \( i \), where \( i = \sqrt{-1} \).

• In the examined polynomial \(3x^4 - 11x^3 - 3x^2 - 6x + 8 = 0\), after identifying real roots, a complex root was found: \( x = 2i \).
• These types of roots are confirmed by failure to discover further real roots, suggesting that the remaining solutions require complex numbers.

While these roots might seem abstract, they are vital in advanced applications like signal processing and quantum mechanics, which rely on complex number theory.