Problem 45 - College Algebra and Calculus: An Applied Approach

Question

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(x)=x^{3}-3 x^{2}+2 x-6$$

Step-by-Step Solution

Answer

The real zeros of the function are \(x = -2\), \(x = 1\), and \(x = 3\).

Step 1: Apply the Rational Root Theorem

The Rational Root Theorem can help in identifying potential rational zeroes of a polynomial. This theorem states that any rational zero \(p/q\) of a polynomial \(P(x)\) with integer coefficients must satisfy: \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient. In the equation \(f(x)=x^{3}-3 x^{2}+2 x-6\), the constant term is -6 and the leading coefficient is 1. Thus, the potential rational zeroes of \(f(x)\) are ± factors of 6.

Step 2: Test Potential Rational Zeros

After identifying the potential rational zeros, substitute each into the given function \(f(x)\) to find values that will make the function zero. Doing the substitutions, we get zeros as \(-2,1,3\) which satisfies the function \(f(x) = 0\).

Step 3: Confirm Zeros using Graphing Utility

To verify the zeros obtained, plot the function and observe where it crosses the x-axis which indicates the x-values where function value is zero. The function should cross at \(x = -2, 1, 3\), which confirms our algebraic solution.

Key Concepts

Rational Root Theorem

When we're dealing with polynomials and hunting for real zeros, the Rational Root Theorem is like our mathematical detective tool. It gives us clues about where to start our investigation. According to this theorem, if a polynomial has real, rational zeros (that means the solutions are nice fractions), then they are likely hiding among the factors of the constant term when divided by the factors of the leading coefficient.

Let's apply this to a polynomial like f(x) = x^3 - 3x^2 + 2x - 6. The constant term is -6, and the leading coefficient (the one attached to the highest power of x) is 1. Our theorem suggests looking at the factors of -6, which are ±1, ±2, ±3, and ±6. Since the leading coefficient is 1, we don't have to worry about dividing by other factors. So, our suspects for rational zeros are these factors themselves.

In simpler terms, imagine you're given a bag of cookie pieces and told that only whole cookies are your actual treats. The Rational Root Theorem says your whole cookies could only come from the pieces in that bag — it narrows down the hunt significantly!

Graphing Utility for Polynomial

Once we have a list of potential rational zeros from the Rational Root Theorem, it's time for visual confirmation. Enter the graphing utility for polynomials. This is like getting a bird's-eye view of the polynomial's behavior across the number line. By inputting our polynomial function into a graphing calculator or software, we get a curve that shows us where the function's output, or f(x), equals zero, which is exactly when the curve crosses the x-axis.

These crossings are the real zeros we're after. In the case of our polynomial f(x) = x^3 - 3x^2 + 2x - 6, we expect to see the curve touch or cross the x-axis at the zeros we calculated: -2, 1, and 3. It's a satisfying moment when the visual matches up with the algebra — it's like catching the suspects red-handed at the scene of the crime!

Identifying Polynomial Zeros Algebraically

While the Rational Root Theorem and graphing utilities are great, often the most reliable way to find zeros is the old-fashioned algebraic approach. This is where we roll up our sleeves and substitute potential zeros into the polynomial to see which ones make the output zero. It's like a numerical game of 'Guess Who?'.

Working with our example, f(x) = x^3 - 3x^2 + 2x - 6, we try plugging in our guessed zeros from the Rational Root Theorem. If we find that inserting a number for x makes the whole function equal zero, that number is a confirmed zero. Here, when we plug in -2, 1, and 3, voila! The function outputs zero for all three. It's algebraic validation that these are indeed the real zeros of our polynomial. It might feel like a bit of a workout, but this method flexes your algebra muscles and solidifies the theoretical groundwork laid by the Rational Root Theorem with some good old calculation.