If f(5) = 0, this indicates that x = 5 is a root of the polynomial function f(x). According to the Remainder Theorem, if you divide a polynomial f(x) by (x-c), the remainder of this division is equal to f(c). Since f(5) = 0, it means that (x – 5) is a factor of the polynomial f(x).
To find all the factors of the function, we need to perform polynomial long division or synthetic division on f(x) using (x – 5) as a divisor. The result will give us a quotient polynomial g(x) such that:
f(x) = (x - 5) * g(x)
The degree of g(x) is one less than the degree of f(x). By further factoring g(x), we can identify more roots and corresponding factors until we fully factor the polynomial function.
For example, if after dividing we find that:
g(x) = x^2 + 3x + 2
we can factor g(x) as:
g(x) = (x + 1)(x + 2)
Thus, the complete factorization of f(x) would then be:
f(x) = (x - 5)(x + 1)(x + 2)
In summary, to find all the factors of the function f(x), we determine that (x – 5) is a factor due to f(5) = 0 and additionally factor the remaining polynomial g(x) obtained from the division process.