Lecture Notes: Degree 4 Polynomial Function in Factored Form

Understanding the Polynomial Function

• We want to find an equation for a degree 4 polynomial function and leave it in factored form.
• A degree 4 polynomial can have at most four real rational zeros or roots.
• The graph shows four x-intercepts, indicating all roots are rational.
• The x-intercepts (roots) are at (-3, -1, 2, 5).

Writing the Polynomial in Factored Form

• Knowing the roots, we can write the polynomial as:
  • ( f(x) = a(x + 3)(x + 1)(x - 2)(x - 5) )
• Here, "a" is a constant to be determined.

Determining the Constant "a"

• We need another point on the graph to find "a".
• Use the y-intercept, which is -15 (point (0, -15)).
• Substitute x = 0 into the function:
  • ( f(0) = a(3)(1)(-2)(-5) = -15 )
• Solving for "a":
  • ( 30a = -15 )
  • ( a = -\frac{1}{2} )

Final Polynomial Function

• The polynomial in factored form is:
  • ( f(x) = -\frac{1}{2}(x + 3)(x + 1)(x - 2)(x - 5) )
• Characteristics:
  • Leading coefficient is negative.
  • Even degree polynomial.
  • Approaches negative infinity as the function moves left and right.

Conclusion

• This completes the process of writing the degree 4 polynomial function in factored form.
• As seen, the graph behavior aligns with expectations for a negative leading coefficient and even degree.