Firstly, let's work on factoring the given polynomial [tex]\( f(x) \)[/tex] and then solving for the roots where [tex]\( f(x) = 0 \)[/tex].
Given:
[tex]\[ f(x) = x^3 - 7x^2 + 7x + 15 \][/tex]
To factor this polynomial, we need to express it as a product of simpler polynomials.
### Factoring the Polynomial
We need to find the roots of the polynomial first, which are the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 0 \)[/tex].
Upon factoring, we find that [tex]\( f(x) \)[/tex] can be expressed as:
[tex]\[ f(x) = (x - 5)(x - 3)(x + 1) \][/tex]
This means the polynomial [tex]\( x^3 - 7x^2 + 7x + 15 \)[/tex] factors into the product [tex]\((x - 5)(x - 3)(x + 1)\)[/tex].
### Solving the Equation [tex]\( f(x) = 0 \)[/tex]
Once we have the factored form, we can solve the equation [tex]\( f(x) = 0 \)[/tex] by setting each factor equal to zero:
[tex]\[ (x - 5)(x - 3)(x + 1) = 0 \][/tex]
The solutions are the roots of the equation:
1. [tex]\( x - 5 = 0 \implies x = 5 \)[/tex]
2. [tex]\( x - 3 = 0 \implies x = 3 \)[/tex]
3. [tex]\( x + 1 = 0 \implies x = -1 \)[/tex]
Therefore, the roots of the polynomial [tex]\( f(x) = x^3 - 7x^2 + 7x + 15 \)[/tex] are:
[tex]\[ x = -1, x = 3, \text{ and } x = 5 \][/tex]
### Summary
- The factored form of the polynomial is:
[tex]\[ f(x) = (x - 5)(x - 3)(x + 1) \][/tex]
- The solutions to the equation [tex]\( f(x) = 0 \)[/tex] are:
[tex]\[ x = -1, x = 3, \text{ and } x = 5 \][/tex]
