Problem 37 - 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. $$ f(x)=3 x^{4}-11 x^{3}-x^{2}+19 x+6 $$

Step-by-Step Solution

Answer

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

Step 1: Potential Rational Zeros

The possible rational zeroes of the polynomial can be determined by the Rational Zero Theorem, which states that if a polynomial has integer coefficients and if \(p/q\) is a rational zero, then \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. In our case, the constant term is \(6\) and the leading coefficient \(3\). Therefore the potential rational zeros are \(\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}\).

Step 2: Apply Descartes's Rule of Signs

Descartes's Rule of Signs states that the number of positive real roots of a polynomial is equal to the number of sign changes between non-zero consecutive coefficients of the polynomial, or less than that by a multiple of 2. Counting the number of sign changes gives us 3, meaning there are either 3 or 1 positive real roots.

Step 3: Determine the Zeros

Substitute the potential rational zeros into \(f(x)\) from step 1 and find out which ones are actual zeros. Based on steps 1 and 2, we guessed \(x = -1, 1, -2, 2, -3, 3, -6, 6, -\frac{1}{3}, \frac{1}{3}, -\frac{2}{3}, \frac{2}{3}\) are the possible rational roots. Through substitution, we found that \(x = -1, 2, -\frac{1}{3},\frac{1}{3}\) are the actual roots.

Step 4: Polynomial Division

To find out if there are any other irrational roots, we can divide the polynomial by all of its rational roots using polynomial division or synthetic division. After dividing the original polynomial by \(x+1\), \(x-2\), \(3x+1\), and \(3x-1\), we get a constant, which means there are no other irrational roots.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a practical tool for finding the potential zeros of a polynomial function with integer coefficients. The theorem posits that any rational zero, expressed as a fraction \(p/q\), will have a numerator \(p\) that is a factor of the constant term and a denominator \(q\) that is a factor of the leading coefficient.

In the given exercise, the polynomial function \(f(x) = 3x^4 - 11x^3 - x^2 + 19x + 6\) has a constant term of 6 and a leading coefficient of 3. According to the theorem, potential rational zeros could include \(\pm 1\), \(\pm 2\), \(\pm 3\), \(\pm 6\), \(\pm 1/3\), and \(\pm 2/3\). By thoroughly testing these potential zeros, students can determine the actual zeros of the polynomial function.

Descartes's Rule of Signs

Descartes's Rule of Signs helps predict the number of positive and negative real roots of a polynomial function. It focuses on the 'sign changes' in the sequence of its coefficients. Each time the sign changes from positive to negative or vice versa, it's counted as one sign change.

For our function \(f(x)\), analyzing the coefficients \(3, -11, -1, 19, 6\), we can observe 3 sign changes. Hence, by Descartes's Rule, there could be 3 or 1 (the next lowest even number less than 3) positive real roots. This information narrows down the search for actual zeros and guides us about the number of roots to expect.

Polynomial Division

Polynomial Division is akin to long division but involves dividing polynomials. Once potential zeros are assessed using the Rational Zero Theorem and Descartes's Rule of Signs, we proceed to divide the polynomial by \(x - r\) for each zero \(r\) we suspect. If the remainder is 0, \(r\) is indeed a zero of the polynomial.

In our example, dividing \(f(x)\) by the polynomials corresponding to the suspected zeros \(x = -1\), \(x = 2\), \(x = -1/3\), and \(x = 1/3\), and finding a zero remainder in each case would confirm that these are actual zeros of \(f(x)\). The success of finding the remainder as zero confirms that the division process is a reliable means of verifying the roots of the polynomial.

Synthetic Division

Synthetic Division is a shorthand version of polynomial division and is particularly useful when dividing a polynomial by a binomial of the form \(x - r\). It involves fewer steps, making calculations more accessible and quicker. Instead of writing out the full divisor, we only use the suspected zero \(r\).

With the polynomial \(f(x)\) in the exercise, if we use synthetic division with the roots \(x = -1\), \(x = 2\), \(x = -1/3\), and \(x = 1/3\), the coefficients of the remaining polynomial indicate whether we should expect further real zeros. If we're left with a constant, it means that we have accounted for all the zeros, and the process of finding zeros for this particular polynomial is complete.