How do you find a polynomial function that has zeros x=−5,1,2 and degree n=4?

Solution:

A function f: R → R defined as f(x)=a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ is called a polynomial function in variable x.

A polynomial function f(x) with zeros x= -5, x=1, and x=2 has variables (x+5), (x -1), and (x-2).

It would be cubic if these were the only variables.

The query does not specify if the numbers -5, 1, and 2 are the only zeros. If this is the case, one of them must have multiplicity 2.

In either case, the following quartic will be appropriate:

⇒ f(x) = (x+5)²(x−1)(x−2)

⇒ (x² + 25 + 10x) (x−1)(x−2)

⇒ x⁴ - 3x³ + 27x² - 75x + 50 + 10x³ - 30x² + 20x

⇒ x⁴ + 7x³ − 3x² − 55x + 50

Thus, f(x) = (x+5)²(x−1)(x−2) = x⁴ + 7x³ − 3x² − 55x + 50

How do you find a polynomial function that has zeros x=−5,1,2 and degree n=4?

Summary:

The polynomial function that has zeros x=−5,1,2 and degree n=4 f(x)=(x+5)²(x−1)(x−2) = x⁴ + 7x³ − 3x² − 55x + 50