Problem 48 - Algebra and Trigonometry

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 obtaining the first zero or the first root. $$ f(x)=2 x^{4}+3 x^{3}-11 x^{2}-9 x+15 $$

Step-by-Step Solution

Answer

The roots of the polynomial function \(f(x) = 2x^{4} + 3x^{3} - 11x^{2} - 9x + 15\) are \(x = 1/2\) and the roots of the cubic equation \(x^3 + 2x^2 - 6x + 15 = 0\).

Step 1: Identify the Rational Zeros

According to the Rational Zero Theorem, if a polynomial has integer coefficients, then every rational zero will have the form \(\pm p/q\) where p is a factor of the constant term and q is a factor of the leading coefficient. For the given equation, the constant term is 15 and the leading coefficient is 2. Hence the potential rational zeros would be \(\pm 1, \pm 3, \pm 5, \pm 15, \pm 1/2, \pm 3/2, \pm 5/2, \pm 15/2\).

Step 2: Use Descartes’ Rule of Signs to find number of positive and negative roots

The number of positive real roots either equals the number of sign variations or is less than the number of sign changes by an even number. In \(f(x)\), the sign changes twice. Thus, by Descartes' Rule, there are either 2 or 0 positive real roots. For negative roots, we substitute \(x\) with \(-x\) in \(f(x)\) and again see the sign changes. It yields \(f(-x) = 2x^{4} - 3x^{3} - 11x^{2} + 9x + 15\). Now, the signs change three times. So there are either 3 or 1 negative real roots.

Step 3: Find the actual roots

Now, try the potential rational zeros and find that \(x=1\) is indeed a root of the equation. When \(x=1\) is substituted into the equation, it will equate to zero. The equation can be factored as \((2x - 1)(x^3 + 2x^2 - 6x + 15) = 0\). So, the roots of the function are \(x = 1/2\) and the roots of \(x^3 + 2x^2 - 6x + 15 = 0\). Solve the cubic equation to find the remaining roots.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a valuable tool to help determine the possible rational zeros of a polynomial function with integer coefficients. It states that if a polynomial has a rational zero, it can be expressed in the form \( \pm \frac{p}{q} \). Here, \( p \) is a factor of the constant term, while \( q \) is a factor of the leading coefficient of the polynomial. For example, consider the polynomial \( f(x) = 2x^4 + 3x^3 - 11x^2 - 9x + 15 \). The constant term here is 15, and the leading coefficient is 2. Hence, the potential rational zeros would be the set of fractions \( \pm 1, \pm 3, \pm 5, \pm 15, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{15}{2} \). By testing these values, one can identify which, if any, are actual zeros of the polynomial. A tested zero will simplify the process of solving the polynomial equation.

Descartes's Rule of Signs

Descartes's Rule of Signs is another useful method for determining the number of positive and negative real roots a polynomial may have. It helps by counting the number of sign changes in the sequence of the polynomial's coefficients. To find the number of positive roots, observe the polynomial \( f(x) \). Count the number of times the signs of the coefficients change. For \( f(x) = 2x^4 + 3x^3 - 11x^2 - 9x + 15 \), there are two sign changes, indicating that the polynomial could have 2 or 0 positive real roots. For the number of negative roots, substitute \( -x \) for \( x \) in \( f(x) \), giving us \( f(-x) = 2x^4 - 3x^3 - 11x^2 + 9x + 15 \). Counting the sign changes here, we find three sign changes, meaning there could be 3 or 1 negative real roots. This rule helps to narrow down the possibilities before testing individual zeros.

Finding Roots of Polynomials

Finding the roots of polynomial equations is typically the main goal when dealing with these functions. The process involves a combination of techniques, beginning with the Rational Zero Theorem and Descartes's Rule of Signs. Next, the potential zeros identified are systematically tested by substitution into the polynomial. For our example, after applying the Rational Zero Theorem and Descartes's Rule of Signs, one practical approach is using synthetic division to further test potential roots. By testing \( x = 1 \), and finding that it equates to zero, we identified it as a root, simplifying our polynomial. This reduction enables easier computation to find additional roots.The process then involves solving the simpler polynomial, often using quadratic formulas or further synthetic division, to find the complete set of roots.

Factorization of Polynomials

Once a zero of the polynomial is identified, the polynomial can be factored around this zero to simplify the further search for remaining roots. When factoring, replace the polynomial with simpler, lower-degree polynomials or known factors. In the given polynomial \( f(x) = 2x^4 + 3x^3 - 11x^2 - 9x + 15 \), we found \( x = 1 \) as a root. This allows us to factor the polynomial into \( (x - 1) \) multiplied by a reduced polynomial \( g(x) = 2x^3 + 4x^2 - 6x + 15 \).Continued factorization involves similar methods, perhaps using synthetic division or direct trial and error, until we fully decompose the polynomial into its irreducible factors. This step can greatly simplify finding all remaining roots of the polynomial function.