Problem 33 - 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. $$ f(x)=x^{4}-2 x^{3}+x^{2}+12 x+8 $$

Step-by-Step Solution

Answer

The zeros of the polynomial function \( f(x) =x^{4}-2 x^{3}+x^{2}+12 x+8 \) are \(x = -1\), \(x = -2\), and \(x = 4\).

Step 1: Compute Possible Rational Zeros

For the polynomial \( f(x) =x^{4}-2 x^{3}+x^{2}+12 x+8 \), using the Rational Zero Theorem, the set of all possible rational zeros are factors of the constant term (+8) divided by factors of the leading coefficient (1). Hence, possible rational zeros are: \(\pm1\), \(\pm2\), \(\pm4\), \(\pm8\).

Step 2: Apply Descartes's Rule of Signs

To find out how many positive and negative real roots there are, apply Descartes's Rule of Signs. Count the number of sign changes in \( f(x) \), which is 2, meaning there are 2 or 0 positive real roots. Replace \(x\) with \(-x\) in \( f(x) \) to get \(f(-x)\), count the number of sign changes in \(f(-x)\), which is 1, indicating there's 1 negative real root.

Step 3: Find the Zeros

Try the possible rational zeros to see which makes the polynomial zero, those values will be our actual zeros. Plugging in the possible rational zeros into our function, \(x = -1\), \(x = -2\), and \(x = 4\) results in 0. Thus, the zeros of the function are -1, -2, and 4.

Key Concepts

Rational Zero Theorem

Understanding the Rational Zero Theorem can be a key step in finding the zeros of a polynomial function. Simply put, it's a handy tool that tells us which numbers to even consider as possible zeros. According to this theorem, if a polynomial has integer coefficients, any rational zero (a fraction where both numerator and denominator are integers) must take the form of a fraction where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient.

In our exercise with the polynomial function
\( f(x) = x^4 - 2x^3 + x^2 + 12x + 8 \),
the constant term is 8 and the leading coefficient is 1 (since the highest power of x, which is x^4, has a coefficient of 1). Using the Rational Zero Theorem, the set of potential rational zeros includes \( \pm1 \) , \( \pm2 \) , \( \pm4 \) , and \( \pm8 \) , which are all the factors of the constant term, divided by the factors of the leading coefficient. These potential zeros are our starting point in the search for the actual zeros of the polynomial.

Descartes's Rule of Signs

When you have a list of potential zeros, it's beneficial to narrow it down further to save time and effort. Descartes's Rule of Signs is an elegant workaround that helps predict the number of positive and negative real roots in a polynomial equation. This rule simply requires you to count the number of times the coefficients change signs in your polynomial function.

For the positive real roots, we count the changes in signs in the original polynomial, \( f(x) \). In our exercise, \( f(x) = x^4 - 2x^3 + x^2 + 12x + 8 \), there are two sign changes, suggesting there could be 2 or 0 positive real roots (since the number of positive real roots is either the number of sign changes, or less than that by an even number).

To determine the possible negative real roots, replace each \( x \) with \( -x \) to get \( f(-x) \), and once again, count the sign changes. In our polynomial, there's one sign change in \( f(-x) \), indicating exactly one negative real root. This doesn't tell us what the roots are, but it does give us a clearer scope for our search.

Polynomial Roots

Finding the roots of a polynomial is akin to solving a detective mystery; you have clues (Rational Zero Theorem and Descartes's Rule of Signs) and a target (the actual zeros). The 'roots' or 'zeros' of a polynomial are the values for which the polynomial is equal to zero. They're crucial for understanding the behavior of the function, particularly, where the graph of the polynomial will intersect the x-axis.

To actually find these zeros, as outlined in the solution to our exercise, we try each possible rational zero that we've identified until the polynomial equals zero. It's a process of plugging in numbers and checking results. For our polynomial function \( f(x) \), the zeros found to satisfy the equation were -1, -2, and 4, meaning that at these points, the graph of the polynomial will touch or cross the x-axis.

Remember, polynomial roots can be both real or complex numbers, but in the context of this exercise, our focus was strictly on identifying the real roots based on the given clues.