Problem 12 - 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. $$f(t)=t^{4}-625$$

Step-by-Step Solution

Answer

The zeros of the function are \(t = 5\), \(t = -5\), \(t = 5i\), and \(t = -5i\), and rewritten as a product of linear factors it becomes \(f(t) = (t - 5)(t + 5)(t - 5i)(t + 5i)\).

Step 1: Set the Polynomial Equal to Zero

In order to find the zeros of the polynomial, it is necessary to set the polynomial equal to zero and solve for \(t\). This gives \(t^{4} - 625 = 0\).

Step 2: Solve the Equation

Rearrange the equation so that it says \(t^{4} = 625\). This can be solved easily by taking the fourth root of each side, yielding four possible solutions \(t = \sqrt[4]{625}\), \(t = -\sqrt[4]{625}\), \(t = \sqrt[4]{625}i\), and \(t = -\sqrt[4]{625}i\). Where \(\sqrt[4]{625} = 5\), therefore our solutions are \(t = 5\), \(t = -5\), \(t = 5i\), and \(t = -5i\).

Step 3: State the Polynomial as a product of linear factors

Finally, based on the obtained zeros, the polynomial \(f(t)\) can be factored as \(f(t) = (t - 5)(t + 5)(t - 5i)(t + 5i)\).

Key Concepts

Finding Zeros

Finding zeros of a polynomial means determining which values make the entire polynomial equal to zero. Essentially, it's about finding the roots of the polynomial equation. For the given polynomial function \( f(t) = t^4 - 625 \), we first set it equal to zero: \( t^4 - 625 = 0 \).
This part is crucial because finding zeros helps us understand where the function touches or crosses the horizontal axis on a graph. The zeros of a function are the future building blocks for writing the polynomial in its linear factors form.
To find the zeros, we solve the equation \( t^4 = 625 \). This step paves the way to uncovering the roots by further breaking down the equation.

Linear Factors

Once the zeros are determined, we can express the polynomial in terms of its linear factors. A linear factor is simply a polynomial of degree one, represented in the form \( (t - a) \), where \( a \) is a root. In our case, after finding the roots, the zeros are \( 5, -5, 5i, \) and \( -5i \).
This leads us to rewrite \( f(t) \) using these linear factors:

• \( (t - 5) \): corresponding to the zero \( t = 5 \)
• \( (t + 5) \): corresponding to the zero \( t = -5 \)
• \( (t - 5i) \): corresponding to the zero \( t = 5i \)
• \( (t + 5i) \): corresponding to the zero \( t = -5i \)

Thus, the polynomial \( f(t) \) becomes \( (t - 5)(t + 5)(t - 5i)(t + 5i) \). Each factor relates to one of the zeros found previously.

Complex Numbers

Complex numbers include a real part and an imaginary part, commonly represented as \( a + bi \), where \( i \) is the imaginary unit with the property \( i^2 = -1 \). In polynomial factorizations, especially when dealing with non-real results, complex numbers come into play.
For the given polynomial \( f(t) = t^4 - 625 \), during the solution process, two of the roots are complex: \( 5i \) and \( -5i \). These are derived from taking the fourth root of a negative value, which naturally involves the imaginary unit \( i \).
Understanding complex numbers is essential in the factorization of polynomials because they help us to express factors even when no real-number solutions exist.

Fourth Roots

The fourth root of a number is one of the four identical factors of the number, which, when multiplied together, give the initial number. Mathematically, if \( x^4 = a \), then \( x = \sqrt[4]{a} \).
In solving our polynomial \( f(t) = t^4 - 625 \), we needed to find the fourth root of 625. The fourth root helps us break down higher-degree polynomials into simpler, linear factors, which is critical in solving such equations.
By finding \( \sqrt[4]{625} = 5 \), we derive the roots: \( t = 5 \), \( t = -5 \), \( t = 5i \), and \( t = -5i \). Each root corresponds to one factor of the polynomial. Understanding how to calculate and use the fourth root is pivotal in finding the zeros and factorizing polynomials completely.