Sure, let's factor the polynomial [tex]\( f(x) = x^3 - 36x^2 + 35 \)[/tex] step by step.
1. Identify the polynomial to be factored:
[tex]\[
f(x) = x^3 - 36x^2 + 35
\][/tex]
2. Look for common factors or possible factorizations.
3. Attempt synthetic division or use the Rational Root Theorem to find potential roots of the polynomial. However, in this context, you will use factoring by grouping or other algebraic methods to solve it directly:
4. Write the polynomial in its factored form:
[tex]\[
f(x) = (x - 1)(x^2 - 35x - 35)
\][/tex]
Here, you can check:
- [tex]\( x - 1 \)[/tex] gives [tex]\( x = 1 \)[/tex] as a root since substituting [tex]\( x = 1 \)[/tex] into the original polynomial [tex]\( x^3 - 36x^2 + 35 \)[/tex] yields 0.
- [tex]\( x^2 - 35x - 35 \)[/tex] is the remaining quadratic factor.
Thus, the complete factored form of [tex]\( f(x) \)[/tex] is:
[tex]\[
(x - 1)(x^2 - 35x - 35)
\][/tex]
This is the factored form of the given polynomial [tex]\( x^3 - 36x^2 + 35 \)[/tex].
