91Ó°ÊÓ

Polynomial functions are a type of mathematical expression that involve variables raised to different powers and combined using addition, subtraction, and multiplication. For example, in the polynomial function given in the exercise, \( f(x) = (x+1)^2 (x-1)^3 (x^2 - 10) \), we have variables raised to different powers. This function is already factorized, meaning it is expressed as a product of simpler factors. Each factor contributes to the overall behavior of the polynomial.

Polynomial functions have several key terms:

• **Degree**: The highest power of the variable in the polynomial. In our example, the degree is 6 (since \((x+1)^2\) contributes 2, \((x-1)^3\) contributes 3, and \(x^2 - 10\) contributes 2+0).
• **Coefficients**: The numerical factors accompanying the variables. Here, the coefficients are all implicitly 1.
• **Roots/Zeros**: Values of \(x\) that make the entire polynomial equal to zero.

Finding the zeros of a polynomial is crucial because these points are where the polynomial intersects the x-axis. If you can write a polynomial in its factorized form, finding the zeros becomes much easier.

Factoring polynomials involves expressing the polynomial as a product of simpler polynomials. This step simplifies solving the polynomial equation, as seen in the provided exercise.

For the polynomial \( f(x) = (x+1)^2 (x-1)^3 (x^2 - 10) \), it's already given in its factorized form. Here’s a breakdown:
\begin{itemize}

**\((x+1)^2\)**: This factor indicates that the polynomial has a root at \(x = -1\) with a multiplicity of 2 because the factor is squared.

**\((x-1)^3\)**: This indicates a root at \(x = 1\) with a multiplicity of 3 because it's cubed.

**\(x^2 - 10\)**: This can be further factorized to \((x - \sqrt{10})(x + \sqrt{10})\), showing that the roots are \(x = \sqrt{10}\) and \(x = -\sqrt{10}\), each with a multiplicity of 1.

Factoring makes it straightforward to set each factor to zero, allowing you to solve for the zeros more easily.

The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. This concept is crucial in understanding the behavior of the polynomial near those roots.

In the exercise, the polynomial \( f(x) = (x+1)^2 (x-1)^3 (x^2 - 10) \) has zeros with differing multiplicities:

• **Zero at \(x = -1\)**: It has a multiplicity of 2. This means the graph of the polynomial touches the x-axis at \(x = -1\) but does not cross it, often creating a parabola-like shape at this root.
• **Zero at \(x = 1\)**: It has a multiplicity of 3. Here, the graph crosses the x-axis at \(x = 1\) and flattens at this point, often creating an 'S' like shape at this root.
• **Zeros at \(x = \sqrt{10}\) and \(x = -\sqrt{10}\)**: Each has multiplicity 1. The graph crosses the x-axis at \(x = \sqrt{10}\) and \(x = -\sqrt{10}\), with the polynomial behaving linearly at these points.

Understanding the multiplicity helps visualize and sketch the polynomial function more accurately.