Problem 46 - 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. $$f(x)=x^{3}-4 x^{2}-25 x+100$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x)=x^3-4x^2-25x+100\) are \(x=4\), \(x=5\), and \(x=-5\).

Step 1: Factor the Polynomial Found in the Function

To find the real zeros of the given function, it's required to work on its polynomial, by factoring it. The cubic polynomial given is \(x^3 - 4x^2 - 25x + 100\). We can factor this into \((x-4)(x^2-25)\), by grouping. Then, further factor \(x^2-25\) into \((x-5)(x+5)\). So, the factored polynomial is \((x-4)(x-5)(x+5)\).

Step 2: Find the Roots of the Polynomial

The roots (or zeros) of the polynomial are the values of \(x\) that we can set to individual factors to make them zero. Set \((x-4)=0\), \((x-5)=0\), and \((x+5)=0\) gives us roots \(x=4\), \(x=5\), and \(x=-5\) respectively, which are the real zeros of the polynomial.

Step 3: Use a Graphing Utility to Plot the Function

The last step involves visualizing the result. By plotting the function \(f(x)=x^3-4x^2-25x+100\) on a graphing utility, we can confirm our solution. We should see that the curve of the function intersects the x-axis at \(x=4\), \(x=5\), and \(x=-5\), confirming these as the real zeros of the function.

Key Concepts

Factoring Polynomials

Factoring polynomials is a technique used to simplify polynomials. It involves breaking down a complex polynomial into simpler, more manageable pieces called factors. In the given problem, the polynomial is a cubic expression, specifically \(x^3 - 4x^2 - 25x + 100\). This expression needs to be factored to find its real zeros.

To factor this cubic polynomial, you can start by grouping terms. We see that the given polynomial can be rewritten as \((x-4)(x^2-25)\). This uses the technique known as factoring by grouping, where terms are rearranged to suggest common factors.

Further breaking down the quadratic part \(x^2-25\) into \((x-5)(x+5)\) uses a special factoring method called the difference of squares. Thus, the fully factored form of the polynomial is \((x-4)(x-5)(x+5)\).

Factoring polynomials is a crucial step in finding zeros, as it transforms the equation and reveals potential solutions more clearly.

Roots of Functions

The roots or zeros of a function are critical points where the function's value equals zero. For polynomial functions, these are the x-values that make the polynomial zero. They are significant because they indicate where the graph of the function intersects the x-axis.

In our case, we identified the roots by solving each factor of the fully factored polynomial \((x-4)(x-5)(x+5)\). For each factor of form \((x-a)\), solving for \(x = a\) helps give us the roots:

• \((x-4)=0\) gives \(x=4\)
• \((x-5)=0\) gives \(x=5\)
• \((x+5)=0\) gives \(x=-5\)

These roots \(x=4\), \(x=5\), and \(x=-5\) represent the x-values where the function touches or crosses the x-axis, confirming them as the real zeros or the roots of the polynomial function.

Graphing Polynomials

Graphing polynomials is a powerful visual tool for understanding the behavior of polynomial functions. By plotting the function, you can see where it intersects the x-axis, indicating the real zeros or roots.

Using a graphing utility, such as a graphing calculator or software, makes this visualization straightforward. For our function \(f(x)=x^3-4x^2-25x+100\), plotting it on a graph should show clear intersections at \(x=4\), \(x=5\), and \(x=-5\). These points are where the polynomial equals zero and are visible where the curve of the polynomial function crosses the x-axis.

Graphing not only confirms the algebraic solutions but also helps in understanding the shape and symmetry of the polynomial. It shows the turning points and how the graph behaves before and after the zeros, offering insights into the overall nature of the function.