A quintic polynomial or quintic function is a type of polynomial that has a degree of 5. It means the highest exponent of its variable, typically denoted by x, is 5. The term ‘Quintic’ is derived from a Latin word meaning ‘fifth.’
Formula
A quintic polynomial can be expressed in the standard form as:
f(x) = ax5 + bx4 + cx3 + dx2 + ex + f
Here,
a, b, c, d, e, and f are constants, and a ≠ 0
Here are a few examples of quintic polynomials:
f(x) = x5 + 3x3 – 9x2 + 5
g(x) = 9x5 + 2x3 – x2 + 5
h(x) = x5 – x4 + 4x – 18
k(x) = x5 + 2
m(x) = 2x5 + x3 – 3x + 7
Quintic functions are commonly used in calculus, hydrodynamics, computer graphics, optics, and spatial analysis.
Degree and Roots
A quintic polynomial (fifth-degree polynomial) has 5 roots or solutions, which may be real or complex numbers. Thus, the polynomial can have up to 5 turning points (inflection points) on its graph.
Graphing
Let us graph the polynomial f(x) = x5 – 5x4 + 5x3 + 5x2 – 6x
Thus, the x-intercepts are (-1, 0), (0, 0), (1, 0), (2, 0), and (3, 0)
Choosing Random Values of x and Completing the Table
Now, we choose random values of x from each interval between the x- and y-intercepts. For each chosen value, we evaluate the function and complete the table by recording the corresponding function values.
x
f(x)
-0.5
3.2813
-0.65
3.6308
1.5
1.4063
1.75
1.1279
2.25
-1.7139
2.5
-3.2813
Determining the End Behavior of the Function
If the leading coefficient is greater than zero, then
as x → -∞, f(x) → -∞
as x → ∞, f(x) → ∞
If the leading coefficient is less than zero, then
as x → -∞, f(x) → ∞
as x → ∞, f(x) → -∞
Now, identifying the sign of the leading coefficient of the given function f(x), we get
a = 1 (> 0)
Thus, as x → -∞, f(x) → -∞, and as x → ∞, f(x) → ∞
Plotting the Points
Thus, the polynomial has 5 roots and 4 distinct extrema (2 local maximums and 2 local minimums).
Finding Derivatives
The derivative of a quintic function always results in a quartic function.
Let us now find the derivative of the polynomial f(x) = x5 – 5x4 + 5x3 + 5x2 – 6x
Differentiating the function with respect to x, we get
Graph the quintic polynomial f(x) = x5 – 5x and answer the following: a) Find the x-intercepts b) Determine the turning points and inflection points c) Identify the end behavior
Solution:
a) X-Intercepts: x5 – 5x = 0 ⇒ x(x4 – 5) = 0 ⇒ x = 0 and x4 – 5 = 0 ⇒ x = 0 and x4 = 5 Solving x4 = 5, we get The real x-intercepts are approximately x = 0, -1.495, 1.495 b) Turning Points or Inflection Points: The real turning points occur at x = 1 Here is an inflection point at x = 0 c) End Behavior: Since the leading term is x5: as x → -∞, f(x) → -∞ as x → ∞, f(x) → ∞
Problem: Identifying QUINTIC BINOMIAL
What is a quintic binomial? Which of the following polynomials is a quintic binomial? a) f(x) = x5 b) g(x) = x5 + x3 – 2x + 3 c) h(x) = x5 + 2 d) m(x) = 5x5 – 3x4 + 7x2 + 15
Solution:
A quintic binomial is a type of polynomial that consists of two terms, and the leading term is of degree 5 (meaning the highest exponent is 5). a) Given, f(x) = x5 is of a single term with the degree 5 Thus, it is not a quintic binomial. b) Given, g(x) = x5 + x3 – 2x + 3 consists of 4 terms with the degree 5 Thus, it is not a quintic binomial. c) Given, h(x) = x5 + 2 is of two terms with the degree 5 Thus, it is a quintic binomial. d) Given, m(x) = 5x5 – 3x4 + 7x2 + 2x – 15 consists of 5 terms with the degree 5 Thus, it is not a quintic binomial.
