Problem 51 - College Algebra and Calculus: An Applied Approach

Question

(a) verify the given factors of \(f(x)\), (b) find the remaining factor of \(f(x)\), (c) use your results to write the complete factorization of \(f(x)\), (d) list all real zeros of \(f\), and (e) confirm your results by using a graphing utility to graph the function. Factors \((x+2),(x-4)\) Function $$f(x)=x^{3}-12 x-16$$

Step-by-Step Solution

Answer

The given factors are verified to be \((x+2)\) and \((x-4)\). The remaining factor after division of \(f(x)\) by these factors is \(x\). Hence the complete factorization of \(f(x)\) is \(x \cdot (x + 2) \cdot (x - 4)\). The real zeros of the function obtained from the factorization are \(x = 0\), \(x = -2\), and \(x = 4\). On graphing, the graph should touch the x-axis at these values, confirming these results.

Step 1: Verification of Given Factors

The given factors are \((x+2)\) and \((x-4)\). This can be verified if when any of these factors are put equal to zero, they return the roots of the function. Solving \((x+2) = 0\) gives \(x = -2\), and solving \((x-4) = 0\) gives \(x = 4\). If these roots satisfy the equation \(f(x) = 0\), then the factors are correct.

Step 2: Finding the Remaining Factor

To find the remaining factor, divide the polynomial \(f(x)\) by the given factors. In this case, \((x+2)\) and \((x-4)\), which forms \((x^2 -2x - 8)\). Simplifying \((\frac{x^{3} -12x -16}{x^2 -2x -8})\) returns \(x\). Hence, \(x\) is the other factor.

Step 3: Complete Factorization of \(f(x)\)

Now, we can write the complete factorization of \(f(x)\) as \(f(x) = x \cdot (x + 2) \cdot (x - 4)\)

Step 4: Find Real Zeros

The real zeros of the function are the solutions to the equation \(f(x) = 0\). From the complete factorization, setting \(f(x) = 0\), gives \(x \cdot (x + 2) \cdot (x - 4) = 0\). This implies that \(x = 0\), \(x = -2\), and \(x = 4\).

Step 5: Confirm Results

Lastly, graph the function \(f(x) = x^{3} -12x -16\) on a graphing calculator. The x-intercepts of the graph correspond to the real zeros of the function. Hence if the graph touches the x-axis at \(x = 0\), \(x = -2\), and \(x = 4\), this confirms the earlier results.

Key Concepts

Real Zeros of Polynomial

Understanding the real zeros of a polynomial is an essential concept in algebra. In simple terms, the real zeros are the values of 'x' that make the polynomial equal to zero. These are also known as the 'roots' or 'solutions' of the polynomial equation. When we factorize a polynomial, we break it down into simpler, more manageable parts that reveal these zeros.

For the given polynomial function, finding the real zeros involves setting the function equal to zero and solving for 'x'. The complete factorization of the function, such as the one derived from the exercise, \(f(x) = x \times (x + 2) \times (x - 4)\), directly points to its zeros. Here, the zeros can be determined by setting each factor equal to zero, which results in \(x = 0\), \(x = -2\), and \(x = 4\). These are the points where the graph of the polynomial will intersect the x-axis on a coordinate plane.

Recognizing the connection between zeros and factors is vital for students, as it helps to quickly identify the behavior of the polynomial without performing extensive algebraic manipulation.

Verification of Polynomial Factors

Verifying the correctness of polynomial factors is a step that cannot be understated in algebraic exercises. By confirming each given factor, students guarantee that they are working with the correct building blocks of the polynomial. The process involves a few techniques, with one being the substitution of the factor's zero into the polynomial and checking if the result is indeed zero.

In our exercise, two factors were given: \(x+2\) and \(x-4\). Verification is simple: when \(x+2=0\), \(x=-2\); when \(x-4=0\), \(x=4\). Substituting these values into the original polynomial should yield zero. If they do, our factors are confirmed as correct. Another method is division; the polynomial \(f(x)\) can be divided by the factors. If the division results in a remainder of zero, the factor is validated. For our function, the long division confirmed the factors and helped reveal the additional factor of 'x', further solidifying our confidence in the factorization approach.

It's crucial for students to understand these techniques, as they provide a systematic way to confirm their initial steps in solving polynomial equations and avoid potential errors moving forward.

Graphing Polynomials

Graphing polynomials is more than just a visual exercise; it's a powerful tool that reveals numerous properties about the function, like end behavior, intercepts, and turning points. By plotting the function on a Cartesian plane, real zeros are made apparent as they are the locations where the graph crosses or touches the x-axis.

For a polynomial such as the one in our problem, \(f(x) = x^3 - 12x - 16\), the graph can confirm the analytic solutions we found. By using a graphing calculator or software, students can visually confirm that the x-intercepts are indeed, \(x = 0\), \(x = -2\), and \(x = 4\).

Graphing can also serve as a reality check for solutions found algebraically. If a solution does not appear as an intercept on the graph, it indicates a potential mistake in the algebraic process. As such, it is always a good practice to graph the polynomial function to verify the consistency of results, which encapsulates the interconnectedness between algebraic and graphical approaches in understanding polynomials.