Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. $$f(x)=x^{4}+10 x^{2}+9$$

Step-by-Step Solution

Answer

The zeros of the given function are \(x=\pm i, \pm 3i\). The polynomial as a product of linear factors is \(f(x) = (x-i)(x+i)(x-3i)(x+3i)\).

Step 1: Identifying Quadratic Form

Rewrite the equation by replacing \(x^{2}\) with \(X\), which gives us a quadratic equation \(X^{2}+10X+9=0\) where \(X=x^{2}\).

Step 2: Factoring the Quadratic Equation

Factorize the equation \(X^{2}+10X+9=0\) into \((X+1)(X+9)=0\).

Step 3: Finding the Roots for X

By equating each factor to zero, we get \(X+1=0\) and \(X+9=0\). Solving these gives us \(X=-1\) and \(X=-9\).

Step 4: Replacing X and Solving for x

From Step 1, remember \(X=x^{2}\). Thus, we'll substitute the factors of \(X\) from Step 3 into \(x^{2}\) to obtain the equations \(x^{2}=-1\) and \(x^{2}=-9\). Solving these, we get \(x=\pm i\) and \(x=\pm 3i\).

Step 5: Writing Polynomial as Product of Linear Factors

To obtain the representation in linear factor form, each root must have its own factor. Hence, the equation could be written as \(f(x) = (x-i)(x+i)(x-3i)(x+3i)\).

Key Concepts

Polynomial Zeros

Understanding polynomial zeros is crucial when working with polynomial functions. A zero, also known as a root, is a value of the variable (often represented as \(x\)) that makes the polynomial equal to zero. In the context of the exercise we were dealing with, the polynomial has the function \(f(x)=x^{4}+10x^{2}+9\). Breaking it down:

• We substitute \(x^{2}\) with \(X\) to simplify our polynomial into a quadratic form \(X^{2} + 10X + 9 = 0\).
• Through factorization, we find the zeros of this quadratic polynomial by solving \((X+1)(X+9)=0\).
• These zeros give us \(X=-1\) and \(X=-9\), which after substituting back for \(x^2\) provide the polynomial zeros \(x=\pm i\) and \(x=\pm 3i\).

By understanding these steps, you're breaking down the equation into bite-sized pieces that reveal the zeros of the polynomial. Each zero corresponds to a point where the polynomial graph touches or crosses the x-axis.

Linear Factors

Writing a polynomial as a product of linear factors is another fundamental aspect of understanding polynomial expressions. Each zero of a polynomial contributes to the development of its linear factors. In simpler terms:

• Linear factors are expressions of the form \((x - r)\), where \(r\) is a root.
• From our exercise, the zeros \(x=\pm i\) and \(x=\pm 3i\) translate into linear factors: \((x-i)(x+i)(x-3i)(x+3i)\).
• This step is crucial as it helps rewrite any polynomial in such a way that reveals all its zeros, highlighting the structure and behavior of the polynomial's graph.

Writing polynomials as products of linear factors is especially useful for both graphing functions and solving polynomials in equations.

Complex Roots

When dealing with polynomial functions, complex roots often come into play, especially when the polynomials have no real roots. Complex roots appear in conjugate pairs, which means if \((a+bi)\) is a root, \((a-bi)\) must also be a root. Let's make this clearer:

• Complex numbers are numbers of the form \(a + bi\), where \(i\) is the imaginary unit \(\sqrt{-1}\).
• In the exercise, the zeros \(x=\pm i\) and \(x=\pm 3i\) are complex roots, meaning the polynomial does not intersect the real axis, and appear as pairs.
• The term "complex" might imply difficulty, but these roots simply add dimension to our polynomial considerations and solutions.

Recognizing complex roots is invaluable in solving polynomial equations because it underscores possible solutions that aren't strictly real. This enriches the scope of functions we can analyze and interact with, offering a complete picture of the polynomial's behavior.