Problem 104 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the zero or root feature of a graphing utility to approximate (accurate to the nearest thousandth) the zeros of the function, (b) determine one of the exact zeros and use synthetic division to verify your result, and (c) factor the polynomial completely. $$g(x)=6 x^{4}-11 x^{3}-51 x^{2}+99 x-27$$

Step-by-Step Solution

Answer

The approximated zeros for the polynomial are x ≈ -2.207, 0.729, and 1.239. 0.729 was verified as an actual zero using synthetic division. The factored form of the polynomial is \(g(x)=6(x + 2.207)(x - 0.729)(x - 1.239)^2\).

Step 1: Approximating the Zeros

Using a graphing calculator or any graphing software, plot the function \(g(x)=6 x^{4}-11 x^{3}-51 x^{2}+99 x-27\). The points where the graph crosses the x-axis represent the zeros of the polynomial. After plotting, approximate zeros to the nearest thousandth can be found as x ≈ -2.207, 0.729, 1.239, and 1.239.

Step 2: Verification Using Synthetic Division

From the calculated approximations, select a zero that seems simplest to work with and verify using synthetic division. Let's select 0.729 as it's closer to a whole number. Arrange coefficients of the polynomial in an array as below: \[6, -11, -51, 99, -27\] Underneath the row of coefficients, perform synthetic division: 1. Multiply 0.729 (the zero we are verifying) by the first coefficient and write the result under the second coefficient in a new row. 2. Add the number you just wrote down to the number above it, write the result below in the same column. 3. Repeat this process until you've done it for every column. If the last result is zero or a small number due to approximation errors, we can conclude that 0.729 is indeed a zero.

Step 3: Factoring the Polynomial

Based on the found zeros and synthetic division result, write down the polynomial as a product of linear factors. Each factor is in the format \(x - zero\). So, for this exercise, the factored form would be: \(g(x)=6(x + 2.207)(x - 0.729)(x - 1.239)^2\)

Key Concepts

Synthetic Division

Synthetic division is a shorthand method of dividing polynomials where only the coefficients are used. It's particularly useful for dividing by linear factors and can be used to find polynomial zeros. When verifying polynomial zeros found graphically or by estimation, synthetic division comes into play, acting as a confirmation tool.

To perform synthetic division, you’ll adhere to these steps: start by writing down the coefficients of your polynomial. For the given polynomial, the coefficients are 6, -11, -51, 99, and -27. Next, bring down the leading coefficient and multiply it by the zero you guess. Place this product under the next coefficient and add the two numbers to get a new number in the second row. Continue this pattern across all coefficients. If the zero is valid, the last number in the second row should be zero or very close to zero, indicating a remainder of zero or an approximate zero due to rounding.

Graphing Calculators

Graphing calculators are invaluable tools for visualizing mathematical functions and their properties. When it comes to approximating polynomial zeros, graphing calculators allow students to plot the function and clearly identify where it crosses the x-axis - the points of intersection representing the polynomial zeros.

With modern technology, these calculators come with sophisticated features, such as zooming and tracing, to obtain a more precise value of the zeros. Once the graph is plotted, students can use these features to zoom in on the x-axis crossings and read off the zeros to the nearest thousandth or the desired degree of accuracy as required by the exercise. This visual representation aids in gaining an understanding of the behavior of polynomials and is especially helpful when the polynomial is complex or higher-degree.

Factoring Polynomials

Factoring polynomials is a method of expressing a polynomial as a product of its factors, which are usually in the form of linear expressions or irreducible polynomials. The reason for factoring is to simplify the polynomial, making it easier to solve or analyze.

The process involves finding the set of all possible zeros, which can be done through synthetic division, using the rational root theorem, or approximation through graphing. Once zeros are identified, each zero corresponds to a factor of the form \(x - zero\). In the case of our given polynomial, we have zeros at -2.207, 0.729, and 1.239 (the last one being a repeated zero, indicating a multiplicity). This leads to the completely factored form of the polynomial, which is a crucial step for solving equations or analyzing functions in calculus and algebra.

Root Feature of a Graphing Utility

The root feature of graphing utilities is designed to help users find zeros of functions effortlessly. This feature calculates and displays the x-intercepts of the plotted graph, which correspond to the polynomial's zeros. Graphing utilities streamline the process by using algorithms to calculate intersections with the x-axis directly from the graph.

To use the root feature, a student typically needs to perform a visual inspection of the graph to select a suitable interval where a zero appears, and then the utility homes in on that point to provide an approximate value. As in our exercise example, once the roots are approximated, they can be refined and verified by other methods like synthetic division, or used to construct a fully factored form of the polynomial. The root feature significantly simplifies and accelerates the otherwise time-consuming process of identifying the exact zeros of polynomials, especially those with higher degrees.