To find all the roots of the function given that one factor is (x – 7), we can apply the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by (x – c), the remainder of that division is f(c). In this case, since we know one of the factors is (x – 7), it indicates that when x = 7, the value of the function f(7) should equal zero, which means that (x – 7) is indeed a factor of f(x).
Now, let’s denote our function as f(x). Since we only know one factor, we can express f(x) as:
f(x) = (x - 7) * g(x)Where g(x) is another polynomial. To find all the roots of f(x), we would need to identify g(x). If we have the complete function definition of f(x), we could factor it completely (if possible) to find the other roots. Without additional information about g(x) or the complete expression for f(x), we can confirm that x = 7 is definitely one root, and the other roots depend on the polynomial g(x).
In summary, while we know that:
- The function f(x) has a root at x = 7.
- To find other roots, we need the explicit form of f(x).