Problem 41 - College Algebra

Question

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. $$ 2 x^{5}+7 x^{4}-18 x^{2}-8 x+8=0 $$

Step-by-Step Solution

Answer

The roots of the given polynomial equation are \(x = -2, -2, -1/2, -1 + \sqrt{3}, -1 - \sqrt{3}\).

Step 1: Apply the Rational Zero Theorem

The Rational Zero Theorem states that any rational root, p/q, of a polynomial must be such that p is a factor of the constant term (in this case 8) and q is a factor of the leading coefficient (in this case 2). Thus, the potential rational roots are: ±1, ±2, ±4, ±8, ±1/2, ±1/4.

Step 2: Apply the Descartes’s Rule of Signs

Descartes's Rule of Signs helps determine the number of positive and negative real roots. Count the number of sign changes in the polynomial. For the given polynomial, there are three sign changes, so there are 3 or 3-2n positive real solutions, which means 3, 1 or none. To find the number of negative solutions, replace x by \(-x\) in the polynomial and repeat the process. The given polynomial becomes \(2x^5 - 7x^4 - 18x^2 + 8x + 8\). There are two sign changes in the polynomial, indicating 2 or 0 negative solutions.

Step 3: Estimate using a graphing utility

Plotting the function on a graphing utility allows for an estimation of the roots. While the exact roots cannot be deduced from this, it helps to validate whether roots are positive or negative and reality check the number of roots. By evaluating the remaining possible roots from step 1, around these estimated values, we can find a root to start synthetic division.

Step 4: Synthetic Division

First, trial and error is used to identify \(x = -2\) as a root. Set up and perform synthetic division using this root. If the remainder is 0, then \(x = -2\) is indeed a root and the quotient will be the reduced form of the polynomial. From here, repeat synthetic division with the remaining roots until the polynomial is fully factored. Then find the roots of polynomial obtained after synthetic division.

Step 5: Find all roots

After applying synthetic division with \(x = -2\), the result is a fourth degree polynomial: \(2x^4 + 3x^3 + 12x^2 + 24x + 4 = 0\). Then, repeating the Rational Zero Theorem, Descartes’s Rule of Signs, and synthetic division again, we can find the remaining roots, \(x = -2, -1/2\). Lastly, the polynomial is factored down to a quadratic equation, and quadratic formula can be used to obtain the two last roots \(x = -1 \pm \sqrt{3}\). Thus, the roots of the equation are \(x = -2, -2, -1/2, -1 + \sqrt{3}, -1 - \sqrt{3}\).

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a useful tool when you need to find the potential rational zeros of a polynomial function. It works by providing a list of possible rational roots that the polynomial may have. This list is generated by taking the factors of the constant term and the factors of the leading coefficient.

• First, identify the constant term of the polynomial, which is the number without any variable attached. For the polynomial given, it is 8.
• Next, identify the leading coefficient, which is the coefficient of the term with the highest degree. In this case, it's 2 for the term with degree 5.
• The possible rational zeros are combinations of the factors of the constant term (8) over the factors of the leading coefficient (2).
• Hence, the potential rational zeros can be: ±1, ±2, ±4, ±8, ±1/2, ±1/4.

After generating the list of possible zeros, each can be tested to see whether it is an actual zero of the polynomial.

Descartes's Rule of Signs

Descartes's Rule of Signs is a mathematical technique used to predict the number of positive and negative real zeros in a polynomial. It operates by analyzing the changes in sign of the polynomial's coefficients.

• To find the number of positive real zeros, you count the number of times the sign changes between consecutive non-zero coefficients of the polynomial. In our polynomial, there are three such sign changes indicating either 3 or 1 positive real zeros.
• To determine the number of negative real zeros, replace every occurrence of the variable, in this case, \(x\), with \(-x\). Then count the sign changes again. For our polynomial, this results in two sign changes meaning there are either 2 or none negative real zeros.

This rule provides a quick insight into the potential distribution of real zeros without solving the entire polynomial.

Synthetic Division

Synthetic division is a simplified process used for dividing polynomials, which is especially handy for evaluating potential zeros from the Rational Zero Theorem.

• This process involves using a suspected zero, often identified from the Rational Zero Theorem or graph estimation, to simplify the polynomial and test if the suspected zero is accurate.
• If the division ends with a remainder of 0, the suspected zero is, indeed, a true zero, and the quotient gives us the reduced polynomial of a lower degree.
• For our example, \(x = -2\) was tested and found to divide the polynomial evenly, indicating that -2 is a root, and the polynomial can then be rewritten in a reduced form.
• This reduced form can be further evaluated by repeating synthetic division, to continue breaking it down until only irreducible parts remain.

Synthetic division considerably simplifies the task of polynomial simplification and factorization.

Quadratic Formula

The Quadratic Formula is a famous formula used to find the roots of a quadratic equation, which is essential when a polynomial is factored down to a quadratic form.

• The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), and its roots can be calculated using the formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
• Once you have factored a polynomial down to a quadratic equation, substitute the respective coefficients into this formula to find the roots.
• In our example, after applying synthetic division multiple times, we end up with a quadratic equation, and employ the quadratic formula to find the remaining roots of the form \(x = -1 \pm \sqrt{3}\).

This highlights the Quadratic Formula's role as a straightforward method of discovering the remaining zeros after reducing polynomials to quadratic form.