Finding Degree 5 Polynomial Equation: Factored Form Analysis

TLDR; Finding a degree 5 polynomial function from the graph, analyzing roots, and determining the equation in factored form.

💡 Finding Equation for Degree 5 Polynomial

The goal is to find an equation for the graph of a degree 5 polynomial function in factored form.

With a degree 5 polynomial function, there are at most five real rational zeros or roots to consider.

Analyzing the graph, it's noted that there are two x-intercepts with different behaviors, indicating the multiplicities of the roots.

📈 Multiplicity Analysis

The root at -2 has an even multiplicity, likely 2, as it touches but does not cross the x-axis.

On the other hand, the root at +2 crosses the x-axis, suggesting an odd multiplicity, likely 3 due to its behavior.

This analysis ensures that there are 5 real rational zeros, crucial for finding the equation in factored form.

🧮 Using Zeros to Find Equation

By knowing the zeros of the function, the equation can be determined in factored form using the given form where roots are denoted as R sub 1, R sub 2, etc., and 'A' is the constant.

To find the value of 'A', another point on the function - the y-intercept - is used, ensuring a comprehensive understanding of the polynomial equation.

🔍 Factorization and Value of A

The factorization of the quadratic function involving the known zeros is carried out, leading to the determination of the value of 'A' using the y-intercept point (0, 32).

This process involves considering the multiplicities of the roots and their corresponding factors, ensuring an accurate representation of the polynomial function.

✅ Writing the Function

With 'A' determined as -1, the equation is written in factored form, incorporating the roots and their respective multiplicities.

The resulting equation is F of X = -1 * (X + 2)^2 * (X - 2)^3, representing the degree 5 polynomial function in factored form.

🔜 Next Example in the Next Video

The next video will feature another example, continuing the exploration of finding equations for polynomial functions from their graphs.

The insightful analysis and application of mathematical concepts will continue in the upcoming content.