To find the roots of the polynomial function with one factor being x – 7, we can apply the Remainder Theorem. According to this theorem, when we divide a polynomial by a linear factor of the form x – c, the remainder of this division is simply the value of the polynomial evaluated at c. In this case, c = 7.
Let’s denote our polynomial as P(x). If x – 7 is a factor of P(x), it means that:
P(7) = 0
This indicates that x = 7 is one of the roots of the polynomial. To find all roots, we would typically need to factor the polynomial or apply numerical methods to find any additional roots that may exist.
For example, if P(x) = x^2 – 14x + 49, we can rewrite this polynomial as:
P(x) = (x – 7)(x – 7) = (x – 7)^2
From this, it’s clear that the root x = 7 has a multiplicity of 2. Thus, the complete root set for this function would simply be:
- x = 7 (with multiplicity 2)
In conclusion, the root of the polynomial with one factor being x – 7 is x = 7, and we can find its multiplicity by further factoring or evaluating the polynomial.