Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$h(x)=x^{2}-4 x-3$$

Step-by-Step Solution

Answer

The zeros of the function are \(x = 2 + \sqrt{7}\) and \(x = 2 - \sqrt{7}\). The polynomial written as a product of linear factors is \(h(x) = (x - (2 + \sqrt{7})) (x - (2 - \sqrt{7}))\).

Step 1: Find the Zeros

To find the zeros of the function, we set it equal to zero and solve for x:\(0 = x^{2} - 4x - 3\). Using the quadratic formula \(x = \frac{ -b \pm \sqrt{b^{2} - 4ac}} {2a}\), with \(a=1, b=-4, c=-3\), we get \(x = 2 \pm \sqrt{7}\). Therefore, the two zeros of the function are \(x = 2 + \sqrt{7}\) and \(x = 2 - \sqrt{7}\).

Step 2: Rewrite as a product of linear factors

A linear factor is of the form (x - a) where 'a' is the zero of the function. Therefore, the function can be written in the form \(h(x) = (x - ( 2 + \sqrt{7})) (x - ( 2 - \sqrt{7}))\). This shows the original polynomial written as product of its linear factors.

Step 3: Verification using the graphing utility

Although it's not possible to show a graphical representation here, this step would involve graphing the original function and checking whether it crosses the x-axis at the zeros identified in step 1. The graph would show that the function does indeed cross at these points, verifying the solution.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool that allows us to find the zeros of any quadratic function of the form \(ax^2 + bx + c = 0\). Solving it directly gives the values of \(x\) by applying the formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic expression. The part under the square root sign \(b^2 - 4ac\) is called the discriminant. It determines the nature of the zeros:

• If the discriminant is positive, there are two distinct real zeros.
• If it is zero, there is exactly one real zero.
• If it is negative, the zeros are complex or imaginary.

In our exercise, substituting \(a = 1\), \(b = -4\), and \(c = -3\) into the quadratic formula provides the zeros \(x = 2 + \sqrt{7}\) and \(x = 2 - \sqrt{7}\). These are real and distinct due to the positive discriminant.

Linear Factors

Once the zeros of a polynomial are found, it can be expressed as a product of linear factors. A linear factor is an expression of the form \((x-a)\), where \(a\) is a root or zero of the polynomial. For the given function \(h(x) = x^2 - 4x - 3\), we found the zeros to be \(x = 2 + \sqrt{7}\) and \(x = 2 - \sqrt{7}\).

Therefore, the linear factors are:

• \((x - (2 + \sqrt{7}))\)
• \((x - (2 - \sqrt{7}))\)

When we multiply these linear factors together, we return to the original polynomial. Expressing a polynomial this way shows its complete factorization, making it easier to understand and analyze its properties. This step is crucial in algebra as it simplifies many calculations related to polynomial functions.

Graphing Utilities

Graphing utilities are tools like graphing calculators and software that allow us to visually inspect functions. By plotting the graph of a polynomial function, we can observe its behavior and verify its zeros.

For our function \(h(x) = x^2 - 4x - 3\), plotting the graph can help confirm the zeros \(x = 2 + \sqrt{7}\) and \(x = 2 - \sqrt{7}\). The graph should intersect the x-axis at these points, indicating that the values are indeed the zeros of the polynomial.

• Ensure the viewing window is set correctly to capture the important features of the graph.
• Check if the intersection points align with calculated zeros.

Using a graphing utility provides a visual representation, which can be extremely helpful in understanding and verifying algebraic solutions. These tools are also useful for exploring variations and deeper insights into different functions.