Problem 47 - 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. $$ f(x)=3 x^{4}-11 x^{3}-x^{2}+19 x+6 $$

Step-by-Step Solution

Answer

The roots of the polynomial \( f(x) = 3x^{4} - 11x^{3} - x^{2} + 19x + 6 \) are -1, -2, 3 and -2/3.

Step 1: Rational Zero Theorem

The Rational Zero Theorem says that if \( f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{1}x + a_{0} \), then every rational zero of \( f(x) \) has the form \( p/q \), where \( p \) is the factor of the constant term (\( a_{0} \)), and \( q \) is the factor of the leading coefficient (\( a_{n} \)). In this equation \( f(x) = 3x^{4} - 11x^{3} - x^{2} + 19x + 6\); the constant term is 6 and the leading coefficient is 3. The factors of 6 are ±1, ±2, ±3, ±6, and the factors of 3 are ±1, ±3. Therefore, the potential rational zeros are ±1, ±2, ±3, ±6, ±1/3, ±2/3.

Step 2: Find Rational Zeros

Substitute each of the potential rational zeros into the equation for \( f(x) \). The numbers that make \( f(x) = 0 \) will be the actual zeros of the function. After testing all the potential zeros, find that -1, -2, and 3 are the zeros of the function. That is, if you substitute -1, -2, or 3 into the equation, it will yield zero. There's one more zero to find.

Step 3: Descartes’s Rule of Signs

Descartes’s Rule of Signs helps to determine the number of positive and negative real roots of a polynomial. The number of positive real roots of \( f(x) \) either is equal to the number of sign changes of \( f(x) \) or is less than that number by a positive even integer. The number of negative real roots of \( f(x) \) is found by applying the rule to \( f(-x) \). The polynomial here in its \( f(x) \) form has two sign changes (3 to -11 and -1 to 19) indicating two positive real roots. The results confirm the positive real root found in the step 2, which was 3. Upon taking \( f(-x) \), two sign changes are observed (3 to 11 and -19 to 6), again indicating two negative real roots. That is consistent with the two negative roots found in step 2, that is -1 and -2.

Step 4: Finding the Last Zero

By synthetic division, divide the original polynomial \(f(x)\) by the factors of the roots you just found (x + 1), (x + 2), and (x - 3). The quotient will tell you the remaining factor, which will give the last root of the polynomial. Upon synthetic division, the remaining polynomial is \(3x+2\), setting this to zero gives a remaining root to be -2/3.

Key Concepts

Rational Zero Theorem

Understanding the Rational Zero Theorem can be a game-changer when you're faced with finding all zeros of a polynomial function. It's like a trusty tool in your math toolbox. Let's say we have a polynomial like the one given in our exercise, with its intimidating powers and coefficients. The Rational Zero Theorem simplifies this by telling us that any potential rational zero is going to be a fraction formed by the divisors of the constant term (that's the number hanging out without an x) over the divisors of the leading coefficient (the number in the front row, glued to the highest power of x).

In the exercise, the constant term is 6, and the leading coefficient is 3. By listing out their factors, we find all possible fractions that could be zeroes of our polynomial. It's essential to consider both positive and negative versions since zeros could be on either side of the number line. After testing these candidates in the polynomial, we discover which ones are truly roots—sort of like picking the right key for a lock!

Descartes's Rule of Signs

Moving on to another fascinating tidbit of algebra is Descartes's Rule of Signs. It's like a forecasting tool for the ups and downs in a polynomial. Here's the trick: by counting how many times the coefficients switch signs, you get a glimpse of how many positive and negative roots to expect. But there's a twist—Descartes tells us that the actual number of positive roots will either match our sign change tally or be less by a multiple of 2.

Looking at our exercise's polynomial, we see two sign changes in the original function, hinting at two potential positive roots, which we confirmed as the root 3. When we feed our polynomial a dose of negativity by plugging in -x, we find the same pattern, suggesting two negative roots, which we nailed down as -1 and -2. It's like a weather prediction that gives us a reliable range, ready for us to pinpoint the exact temperature.

Synthetic Division

Picture synthetic division as an efficient, no-fuss method of dividing polynomials—almost like speed-dating mathematics style. Instead of the long-winded traditional division, synthetic division gets straight to the point, using only the coefficients and discarding the variables and powers for most of the process.

In our exercise, we use this snappy technique to divide the original polynomial by those zeroes we found earlier. It's a systematic process: we jot down our coefficients, then bring down, multiply, and add our way through the chart until we reach the end. The result is a neat row of numbers which translates back into a simpler polynomial. This new polynomial holds the clue to finding our remaining mystery zero, and when we solve for it, we unveil the elusive -2/3. A real 'voilà' moment in our zero-hunting adventure!

Polynomial Roots

Identifying the roots of a polynomial can feel like solving a mystery, where each root is a piece of the puzzle. Roots, or zeros, are the solutions to the equation when we set the polynomial equal to zero. They're the exact points where the polynomial graph kisses the x-axis, giving us the full picture of where the graph lives and breathes.

In our exercise's context, we rolled up our sleeves to uncover all roots of a quartic polynomial. By leveraging the Rational Zero Theorem, we found a lineup of suspects. With Descartes's Rule, we predicted how many positive and negative witness sightings to expect. Then, synthetic division swooped in, helping us confirm these sightings and track down the last incognito number. Together, these roots tell the complete narrative of our polynomial's journey across the coordinate grid, providing invaluable insight into its nature and behavior.