Question

Determine the number of zeros of the polynomial function. $$f(t)=-2 t^{5}-3 t^{3}+1$$

Short answer

The polynomial function \(f(t)=-2 t^{5}-3 t^{3}+1\) has five zeros.

Step-by-step solution

Identify the Degree of the Polynomial

The degree of the polynomial function \(f(t)=-2 t^{5}-3 t^{3}+1\) is 5, which is the highest power of the variable \(t\).

Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) complex roots. Therefore, this polynomial will have exactly 5 roots (or zeros), including repeated roots and complex roots (those involving imaginary numbers).

Key concepts

Degree of a Polynomial

The degree of a polynomial is a crucial aspect when understanding polynomial functions. It refers to the highest power of the variable with a non-zero coefficient in a polynomial expression. For the polynomial function \(f(t) = -2t^5 - 3t^3 + 1\), the degree is 5 because the term with the highest exponent of the variable \(t\) is \(-2t^5\). This degree tells us a lot about the shape and behavior of the polynomial's graph.

A few key points about the degree of a polynomial:

• The degree indicates the maximum number of zeros or roots the polynomial can have, although they might not all be distinct or real.
• The degree dictates the end behavior of the graph. For example, odd-degree polynomials (like our example of degree 5) will have opposite end behaviors (as \(t\) approaches \(\pm\infty\)).
• In general, it determines the general number of turning points or changes in direction the graph can have, which is the degree minus one.

Understanding the degree of a polynomial gives insight into its complexity and the nature of its solutions.

Zeros of a Polynomial

Zeros or roots of a polynomial are the solutions of the equation when the polynomial is set to zero. These are the values of the variable that make the polynomial equal to zero. For the polynomial \(f(t) = -2t^5 - 3t^3 + 1\), the zeros are the solutions to the equation \(-2t^5 - 3t^3 + 1 = 0\).

Key points to remember about zeros of polynomials:

• A polynomial of degree \(n\) can have up to \(n\) zeros, including multiplicities.
• Zeros can be real or complex. Real zeros are where the graph crosses or touches the x-axis, while complex zeros (involving imaginary numbers) do not have intersections with the x-axis.
• In the context of polynomial graphs, finding zeros helps determine important features like intercepts, and can provide clues about general graph shape and direction.

Finding the zeros is key to understanding the full behavior and characteristics of the polynomial.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a cornerstone concept concerning polynomial functions. This theorem states that every non-zero single-variable polynomial equation of degree \(n\) has exactly \(n\) complex roots. These roots may include real numbers and non-real complex numbers, and they count multiplicities.

Here's what the Fundamental Theorem of Algebra means for you:

• It guarantees that polynomial equations have roots in the complex number system, providing a comprehensive framework for understanding solutions.
• Regardless of the polynomial's degree, it reassures us that a solution set exists, even if not all are real or distinct.
• For the polynomial \(f(t) = -2t^5 - 3t^3 + 1\), it assures us there are exactly 5 roots, consistent with its degree.

Understanding and applying the Fundamental Theorem of Algebra allows us to anticipate the number of solutions and informs more complex analysis of the polynomial's nature and behavior.