Problem 40 - 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. $$f(x)=x^{4}-x^{3}-20 x^{2}$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x)=x^{4}-x^{3}-20 x^{2}\) are \(x = 0, x = 5, and x = -4\).

Step 1 - Simplification of the equation

We first simplify the function. The function can be simplified by factoring out \(x^{2}\) which is common in all terms of the equation. Doing so gives us: \[f(x) = x^{2}(x^{2}-x-20)\].

Step 2 - Further simplification and solution

The quadratic equation \(x^{2}-x-20\) is further simplified by using factorization for quadratic equations to avoid quadratic formula. This will resolve into \((x-5)(x+4)\). So, the function becomes: \[f(x) = x^{2}(x-5)(x+4)\]. The zeroes of the function are obtained when \(f(x) = 0\), so setting \(x^{2}(x-5)(x+4) = 0\). The roots of the equation are: \(x = 0, x = 5, and x = -4\).

Step 3 - Graphical Confirmation

To double-check the solution, graph the function \(f(x)=x^{4}-x^{3}-20 x^{2}\) using a graphing calculator or tool. The x-intercepts on the graph also indicate the zeroes of the function. Observe if the graph crosses the x-axis at \(x = 0, x = 5, and x = -4\). If it does, it validates the result obtained in the previous step.

Key Concepts

Polynomial Factorization

Factorization is a key technique in algebra to break down complex polynomials into simpler components, making them easier to work with. This method involves expressing a polynomial as a product of its factors.

In the original exercise, the polynomial given was of the form: \[f(x)=x^{4}-x^{3}-20 x^{2}\].
To simplify this, factor out the greatest common factor (GCF) from all terms. In this case, it's \(x^2\). Factoring \(x^2\) from each term results in:
\[f(x)=x^2(x^2-x-20)\].

The next step is to further factor the quadratic \(x^2-x-20\). This can be done by finding two numbers that multiply to \(-20\) and add to \(-1\) (the coefficient of \(x\)).

• The factors of -20 are: \(-1, 20\), \(1, -20\), \(-2, 10\), \(2, -10\), \(-4, 5\), \(4, -5\).

The correct pair here is \(-5\) and \(4\), because:

• \(-5 + 4 = -1\) (matches coefficient of \(x\))
• and \(-5 \times 4 = -20\)

Thus, \(x^2-x-20\) can be expressed as \((x-5)(x+4)\), putting it all together:
\[f(x) = x^2(x-5)(x+4)\].

This factorization process reveals much about the underlying structure of the polynomial, paving the way to find its zeros.

Real Zeros

Real zeros of a function are the values of \(x\) where the function evaluates to zero. They represent the x-intercepts on a graph, where the curve crosses the x-axis. These points help understand where a function changes direction and they are crucial for analyzing polynomial behavior.

For the polynomial \(f(x) = x^2(x-5)(x+4)\), to find the real zeros, set each factor to zero and solve for \(x\).

• From \(x^2 = 0\): \(x = 0\).
• From \(x-5 = 0\): \(x = 5\).
• From \(x+4 = 0\): \(x = -4\).

Hence, the zeros of the function are \(x = 0\), \(x = 5\), and \(x = -4\).

These zeros suggest the function's x-intercepts, showing distinct points where it touches or crosses the x-axis, helping to visualize and confirm polynomial behavior.

Graphical Confirmation

Once the zeros of the polynomial are algebraically determined, it is essential to confirm these calculations using a graphical approach.

A graphing calculator or a software like Desmos or GeoGebra can be used to plot the function \(f(x) = x^4-x^3-20x^2\). When plotted, look for the points where the curve intersects the x-axis.

• These intersections will occur precisely at the real zeros: \(x = 0\), \(x = 5\), and \(x = -4\).

The graphical representation of the polynomial visually confirms that the algebraic process is correct. Seeing the curve cross the x-axis at these specific points provides a visual assurance that can enhance understanding and instill confidence in the factorization and zero-finding processes. This step helps bridge the gap between algebraic manipulations and their real-world graphical counterparts.