Problem 56 - 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 \((x-\sqrt{2}),(x+2)\) Function $$f(x)=x^{3}+2 x^{2}-2 x-4$$

Step-by-Step Solution

Answer

Given factors are confirmed, and the remaining factor is \(x-2\). The factorization of the function is \(f(x) = (x-\sqrt{2})(x+2)(x-2)\). The real zeros are \(-2\), \(2\), and \(\sqrt{2}\). The graph of the function confirms these results.

Step 1: Verify the Given Factors

Use the factor theorem to check that the given expressions, \(x-\sqrt{2}\) and \(x+2\), are indeed factors of the function \(f(x)\). The factor theorem states that \(x-a\) is a factor of f(x) iff \(f(a) = 0\). So plug \(\sqrt{2}\) and \(-2\) into the function and confirm that they both result in zero.

Step 2: Find the Remaining Factor

Divide the function \(f(x)\) by the product of confirmed factors \((x-\sqrt{2})(x+2)\) to obtain the remaining factor. Though long division or synthetic division could be used, synthetic division is often simpler when dividing polynomials.

Step 3: Complete Factorization of the Function

Now, multiply the confirmed factors and the remaining factor found in the previous step together to form the factorization of the function \(f(x)\).

Step 4: Listing All Real Zeros

The real zeros of the function \(f(x)\) are the x-values for which \(f(x) = 0\) or where the graph of the function intersects the x-axis. They can be found from the factors of the function and are the values that make each factor equal to zero.

Step 5: Confirm Results with a Graphing Utility

Use a graphing utility to plot the function \(f(x)\). The graph should intersect the x-axis at the real zeros found in the previous step, confirming the results obtained. Visual inspection also allows confirmation of the factorization: the graph should have shape and turning points that align with a cubic function that has been factored as found.

Key Concepts

Factor Theorem

Understanding the Factor Theorem is crucial for simplifying polynomial functions and finding their zeros. In essence, this theorem states that for any polynomial function, such as our provided example function f(x), if a certain number a satisfies the equation f(a) = 0, then it's clear that (x - a) is a factor of the polynomial.

Applying it to our exercise, we can verify that (x - \(\sqrt{2}\)) and (x + 2) are factors of the cubic function f(x) = x^3 + 2x^2 - 2x - 4. By substituting \(\sqrt{2}\) and -2 for x in the function and obtaining zero, we confirm that these expressions are indeed factors. It's a simple yet powerful tool that provides a bridge from algebraic expressions to graphical interpretations of polynomial functions.

Synthetic Division

Synthetic division is a shortcut technique, simplifying long division when dividing polynomials, especially handy when dealing with linear divisors of the form (x - c). It reduces the complexity and the number of steps required when compared to the traditional long division method.

In the given exercise, synthetic division would be used after confirming that (x-\(\sqrt{2}\)) and (x+2) are factors of f(x). This process would be instrumental in finding the remaining factor of the cubic function. The method arranges coefficients and employs a series of multiplication and addition operations to reveal the quotient, which in this case would be the other factor we're looking for.

Real Zeros

Real zeros of a polynomial function are the x-values where the function crosses the x-axis. In other words, these are the values of x for which the function’s output f(x) is zero. For the cubic function in our exercise, the real zeros are directly related to the factors (x-\(\sqrt{2}\)) and (x+2).

By setting these factors to zero, we can solve for the x-values which give the real zeros: \(x = \sqrt{2}\) and \(x = -2\). Any additional real zeros can be determined through the same process, by setting the remaining factor equal to zero after division. Identifying the real zeros is valuable for graphing the function and understanding its behavior in different domains.

Cubic Functions

Cubic functions are a type of polynomial function where the highest exponent of the variable x is three. These functions are known for their 'S'-shaped curves, featuring at least one and up to three real zeros, and possibly one or two turning points. In our exercise, the cubic function addressed is f(x) = x^3 + 2x^2 - 2x - 4.

A complete factorization of this function would reveal all its intercepts with the x-axis, granting us profound insights on its graphical representation. Typically, a cubic function intersects the x-axis at least once because it is guaranteed to have at least one real zero, which resonates with the fundamental theorem of algebra stating that every non-zero, single-variable, degree-n polynomial with complex coefficients has, counted with multiplicity, exactly n roots.