Since $f(1) = 0$, we know that $x - 1$ is a factor of $f(x) = x^3 - 4x^2 - 7x + 10$.

To factor $f(x)$, we can use synthetic division to divide $f(x)$ by $x - 1$:

1. Set up the coefficients of $f(x)$: $1, -4, -7, 10$.

2. Perform synthetic division with $x - 1$:

   • Bring down the 1.
   • Multiply 1 by 1 to get 1. Add to -4 to get -3.
   • Multiply -3 by 1 to get -3. Add to -7 to get -10.
   • Multiply -10 by 1 to get -10. Add to 10 to get 0, confirming no remainder.

The result of synthetic division is $x^2 - 3x - 10$, so: $f(x) = (x - 1)(x^2 - 3x - 10)$

3. Factor $x^2 - 3x - 10$:

   • We need factors of -10 that add up to -3, which are -5 and +2.
   • Thus, $x^2 - 3x - 10 = (x - 5)(x + 2)$.

4. Write the fully factored form of $f(x)$: $f(x) = (x - 1)(x - 5)(x + 2)$

Final Answer:

$f(x) = (x - 1)(x - 5)(x + 2)$

Would you like further explanation or have any questions?

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Here are five related questions for further practice:

1. How would you factor a polynomial if the given point does not make $f(x) = 0$?
2. Can synthetic division be used for any divisor other than $x - a$ for a polynomial?
3. How would you verify the factors found for a polynomial?
4. What is the role of the Remainder Theorem in polynomial factorization?
5. How does knowing one root help in finding the remaining roots?

Tip: Always verify each factor by substitution or re-expansion to ensure correctness.