Problem 27 - 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.) $$f(x)=x^{4}+10 x^{2}+9$$

Step-by-Step Solution

Answer

The zeros of the given polynomial function are the solutions obtained from the quadratic equation solved in step 1. The polynomial can be written as a product of linear factors as found in step 2. Verification of this result would be performed using a graphing utility as described in step 3.

Step 1: Find the Zeros

To find the zeros, we set the function equal to zero and solve for \(x\). So, solve the equation \(x^{4} + 10x^{2} + 9 = 0\). Notice that the equation is a quadratic in terms of \(x^{2}\), thus we can use the quadratic formula to solve for \(x^{2}\). The quadratic formula is \((x^{2})=(-b\pm \sqrt{b^{2}-4ac})/2a\), where \(a=1\), \(b=10\), and \(c=9\). Now solve for \(x^{2}\).

Step 2: Convert into Linear Factors

Once we have the solutions for \(x^{2}\), implement the square root of each solution to find the value of \(x\). Since there may be positive and negative square roots, we could have four solutions. Thus, \(f(x)\) can be written as \(f(x) = A(x-x_{1})(x-x_{2})(x-x_{3})(x-x_{4})\), where \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\) are the solutions for \(x\).

Step 3: Verify Graphically

Finally, we should verify the results graphically to ensure accurate solutions. Use a graphing utility to graph \(y = f(x)\). The graph should touch the x-axis at the zeroes found in step 1, verifying their correctness.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It's especially useful when factoring is difficult or impossible. The formula itself is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where:

• \(a\), \(b\), and \(c\) are coefficients from the equation.
• \(b^2 - 4ac\) is known as the discriminant and determines the nature of the roots.

In our exercise, the function \(f(x) = x^4 + 10x^2 + 9\) takes on a quadratic form by setting \(x^2 = y\), which transform it into \(y^2 + 10y + 9 = 0\). Here, \(a = 1\), \(b = 10\), and \(c = 9\). This allows us to apply the quadratic formula to find the values of \(y\), bearing in mind that \(y = x^2\), which leads to the further step of extracting \(x\). Calculating the discriminant helps us anticipate whether we'll have real or imaginary solutions.

Zeros of a Function

Zeros of a function are the points where the function touches or crosses the x-axis on a graph. In other words, they are the solutions to the equation when the function is set to zero, such as finding \(x\) in \(f(x) = 0\). Identifying these zeros is crucial for understanding the behavior of the function. For our specific function \(f(x) = x^4 + 10x^2 + 9\), we find the zeros by setting it to zero and using the quadratic formula to solve \(x^2 + 10x + 9 = 0\). After solving, we get values of \(x^2\) which we then take the square root of to find the actual values of \(x\). Finding these zeros gives us insight into the function's structure and allows us to re-express it using its linear factors.

Graphing Utility

A graphing utility is a useful device or software application that aids in visualizing mathematical equations, making it easier to understand their characteristics and behaviors. When we graph a polynomial function like \(f(x) = x^4 + 10x^2 + 9\), our goal is to graphically confirm its zeros. These zeros are the points where the graph meets the x-axis. With a graphing utility, you can plot \(y = f(x)\) and quickly see if the calculated zeros align with this intersection, confirming their accuracy. Additionally, a graphing utility can provide insights into the function's shape and symmetry, as well as help verify imaginary solutions by supporting complex plane graphing.

Linear Factors

Once potential zeros are found, a polynomial can be expressed as a product of linear factors. Linear factors consist of expressions like \((x - x_1)\) where \(x_1\) is a zero of the function. When our polynomial is rewritten in this form, it showcases the zeros clearly. For function \(f(x) = x^4 + 10x^2 + 9\), after solving \(x^2 = y\) and finding \(x\), we use the square roots of these solutions. The polynomial can then be expressed as a product of linear factors. Understanding linear factors enhances comprehension of the polynomial's zeros, offering a neat multiplication expression to visualize how the polynomial behaves at each intercept.