A polynomial function with a degree of 6 has the zeros of 𝑥 = 2 (order 2), 𝑥 = −1 (order 1) and 𝑥 = 5
(order 3). Write a general formula for a family of functions that share the same properties as this function.
Please HELP!
f(x) = (x + a) (x + b)² (x + c)³
To find a general formula for a family of functions with the given properties, we'll break down the process step by step.
Step 1: Determine the factors of the polynomial
We are given the zeros of the polynomial and their respective orders:
𝑥 = 2 (order 2)
𝑥 = -1 (order 1)
𝑥 = 5 (order 3)
Based on this information, we can deduce the factors of the polynomial. Each zero and its order will contribute a corresponding factor.
For 𝑥 = 2 (order 2), the factor is (𝑥 - 2)^2.
For 𝑥 = -1 (order 1), the factor is (𝑥 + 1).
For 𝑥 = 5 (order 3), the factor is (𝑥 - 5)^3.
Now we can write the polynomial in factored form by multiplying these factors together:
𝑝(𝑥) = (𝑥 - 2)^2(𝑥 + 1)(𝑥 - 5)^3
Step 2: Determine the degree of the polynomial
The degree of the polynomial is determined by the highest exponent in the factored form. Since the exponents for (𝑥 - 2)^2 and (𝑥 - 5)^3 are 2 and 3 respectively, the degree of the polynomial is:
Degree = 2 + 1 + 3 = 6
Step 3: Generalize the formula
To write a general formula for a family of functions with the same properties, we replace the constants in the factored form with variables to create a flexible equation. Let's use "a", "b", "c", and "d" to represent these constants:
𝑝(𝑥) = (𝑥 - a)^2(𝑥 + b)(𝑥 - c)^3
Now, this formula can generate a family of functions that share the same properties as the given polynomial.
For example, if we want a function with the zeros 𝑥 = 2 (order 2), 𝑥 = -1 (order 1), and 𝑥 = 5 (order 3), we can set the values of "a", "b", and "c" in the general formula to match:
𝑝(𝑥) = (𝑥 - 2)^2(𝑥 + 1)(𝑥 - 5)^3
You can plug in different values for "a", "b", and "c" to create various functions within this family.