To factor the polynomial [tex]\( f(x) = x^4 - 10x^3 + 24x^2 + 32x - 128 \)[/tex] completely, follow these steps:
### Step-by-Step Factorization
1. Identify the polynomial:
[tex]\[ f(x) = x^4 - 10x^3 + 24x^2 + 32x - 128 \][/tex]
2. Find potential factors:
We are looking for factors of the polynomial. By inspection or trial and error, we can find that [tex]\( x - 4 \)[/tex] is a factor.
3. Divide the polynomial:
Once we have a factor, we can use polynomial division or synthetic division to divide [tex]\( f(x) \)[/tex] by [tex]\( x - 4 \)[/tex]. Repeated division will show that [tex]\( x - 4 \)[/tex] is a factor multiple times.
4. Factor out the repeated factor:
After dividing [tex]\( f(x) \)[/tex] multiple times by [tex]\( x - 4 \)[/tex], we find that [tex]\( x - 4 \)[/tex] is a factor three times. This gives us the factor [tex]\( (x - 4)^3 \)[/tex].
5. Find the remaining factor:
Upon dividing [tex]\( f(x) \)[/tex] by [tex]\((x - 4)^3\)[/tex], the remaining factor is [tex]\( x + 2 \)[/tex].
Therefore, the complete factorization of [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x - 4)^3 (x + 2)
\][/tex]
### Solving the Equation [tex]\( f(x) = 0 \)[/tex]
With the factorization complete, solving the equation [tex]\( f(x) = 0 \)[/tex] involves finding the roots of the factored polynomial:
[tex]\[
(x - 4)^3 (x + 2) = 0
\][/tex]
This equation is satisfied when either factor equals zero.
1. Set each factor equal to zero:
For [tex]\( (x - 4)^3 = 0 \)[/tex]:
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
For [tex]\( (x + 2) = 0 \)[/tex]:
[tex]\[
x + 2 = 0 \implies x = -2
\][/tex]
2. List the solutions:
The solutions are [tex]\( x = 4 \)[/tex] and [tex]\( x = -2 \)[/tex].
### Final Answer
The factorization of [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x - 4)^3 (x + 2)
\][/tex]
The solutions to the equation [tex]\( f(x) = 0 \)[/tex] are:
[tex]\[
x = -2 \quad \text{and} \quad x = 4
\][/tex]
