Finding Degree 4 Polynomial Function in Factored Form

TLDR; The video explains finding a degree 4 polynomial function from its graph and leaves it in factored form.

💡 Finding the Equation

The goal is to find an equation for a degree 4 polynomial function from its graph and leave it in factored form.

The degree 4 polynomial function has at most four real rational zeros or roots, which are evident from the graph with four x-intercepts.

The x-intercepts (-3, -1) and (+2, +5) represent the roots of the polynomial function, allowing the function to be written in factored form using these roots.

📈 Determining the Value of "A"

To determine the value of the constant "A", the y-intercept (-15) is used to find another point of the function (0, -15).

The process involves finding the factors of the polynomial function from the zeros of the function and then using the point (0, -15) to determine the value of "A".

The factors of the polynomial function are derived from the zeros of the function, and the opposite sign of the zeros from the graph is used to construct the factors.

🔍 Finding the Value of "A"

The value of "A" is found by substituting the point (0, -15) into the function and solving for "A".

By substituting zero for x and setting the function value equal to -15, the value of "A" is calculated to be -1/2.

📝 Writing the Polynomial Function

With the value of "A" (-1/2) known, the polynomial function is written in factored form using this value and the roots.

The resulting equation is F(x) = -1/2(x + 3)(x + 1)(x - 2)(x - 5).

The leading coefficient is negative, and the function is of degree 4, indicating a downward trend as it moves to the left and right, as expected with an even degree and a negative leading coefficient.

🔚 Conclusion

The video concludes with a brief explanation of the behavior of the function as it approaches negative infinity in both directions due to the even degree and the negative leading coefficient.

The viewer is encouraged to look forward to another example in the next video.