Problem 55 - 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 \((x-\sqrt{3}),(x+2)\) Function $$f(x)=x^{3}+2 x^{2}-3 x-6$$

Step-by-Step Solution

Answer

After verifying the given factors and finding the remaining one by polynomial division, the complete factorization of \(f(x)\) can be written. By setting each factor equal to zero, the real zeros of \(f\) are obtained, and the graphical plotting further confirms these roots.

Step 1: Verify Given Factors

As all real zeros of the polynomial are roots of the equation \(f(x) = 0\), let's substitute the roots \(\sqrt{3}\) and \(-2\) from the given factors \((x-\sqrt{3}),(x+2)\) into the equation \(f(x)=x^{3} +2x^{2} -3x -6\). If we get zero for both values, the factors are verified.

Step 2: Find the Remaining Factor

Since we know two of the factors of our cubic polynomial function, we can find the remaining factor by polynomial division. We divide our given cubic function \(f(x)=x^{3} +2x^{2} -3x -6\) by the product of given factors \((x-\sqrt{3})(x+2)\). The quotient will be the remaining factor.

Step 3: Write the complete factorization

Now we write down the complete factorization by multiplying the given factors with the remaining factor found in step 2.

Step 4: List all Real Zeros

Every factor of a polynomial corresponds to a real zero when the polynomial is set equal to zero. Solve the equation \(f(x) = 0\) by setting each factor found in step 3 equal to zero.

Step 5: Confirm Your Results From Graph

A real zero corresponds to an x-intercept of the function's graph. To confirm our results, graph the given function using a graphing utility, and observe if the x-intercepts matches the real zeros found in step 4.

Key Concepts

Real Zeros of a Polynomial

Understanding the real zeros of a polynomial is crucial in analyzing its behavior. Real zeros are the x-values where the polynomial equals zero, in other words, where it intersects the x-axis. Finding these helps break down complex polynomials into simpler factors. For the exercise with the polynomial function f(x) = x^3 + 2x^2 - 3x - 6, the real zeros are the values of x for which f(x) equals zero.

The given factors, (x - \(sqrt{3}\)) and (x + 2), suggest that \(\sqrt{3}\) and -2 are potential real zeros since factors translate to zeros of the polynomial. Verifying them is the first step, which involves substituting these values into the original polynomial. If the output is zero, then they are indeed real zeros. Remember, each real zero corresponds to an x-intercept on a graph of f(x). The real zeros also provide an invaluable shortcut in factoring the polynomial, simplifying the process considerably.

Polynomial Division

Polynomial division, much like long division for numbers, is a method to simplify polynomials and find their factors. In our example, we are given a cubic polynomial and two of its factors. The objective is to find any remaining factors, and this is where polynomial division comes into play.

To find the remaining factor of f(x), divide the polynomial by the product of the given factors, (x - \(sqrt{3}\))(x + 2). The result of this division is the missing factor. If done correctly, there will be no remainder, and the quotient will be a polynomial of lesser degree. This remaining factor will also lead us to another real zero, further unraveling the structure of the polynomial.

Graphing Polynomials

Graphing polynomials is a visual approach to understanding their properties. A polynomial graph can be used to confirm the real zeros and the overall shape of the function. When graphing the function f(x) = x^3 + 2x^2 - 3x - 6, each zero found through algebraic methods should correspond to an x-intercept on the graph. It's particularly helpful to use a graphing utility to see where the polynomial crosses the x-axis.

Moreover, the graph can show the behavior of the polynomial between zeros and at the extremities, revealing where the function is positive or negative, and illustrating local maxima and minima. Observing the end behavior also tells us about the leading coefficients, confirming whether the original factorization is correct. Graphing is not only a confirmation tool but also provides insight that could lead to discovering additional patterns or properties within the polynomial function.