To factor the given cubic polynomial $x^3 - 8x^2 + 17x - 10$, we can use the Factor Theorem to find potential factors.

Step 1: Use the Factor Theorem

The Factor Theorem states that if $x = a$ is a root of the polynomial, then $(x - a)$ is a factor. Let's check for simple integer roots by substituting values of $x$ into the polynomial $f(x) = x^3 - 8x^2 + 17x - 10$.

We'll test small integers like $\pm 1, \pm 2, \pm 5$, which are reasonable guesses based on the constant term $-10$.

Test $x = 1$:

$f(1) = 1^3 - 8(1^2) + 17(1) - 10 = 1 - 8 + 17 - 10 = 0$ So, $x = 1$ is a root, and therefore $(x - 1)$ is a factor.

Step 2: Perform synthetic division

Now that we know one factor is $x - 1$, we can divide the cubic polynomial $x^3 - 8x^2 + 17x - 10$ by $x - 1$ to find the quotient.

Using synthetic division:

1 & 1 & -8 & 17 & -10 \\ & & 1 & -7 & 10 \\ \hline & 1 & -7 & 10 & 0 \end{array}$$ The quotient is $$x^2 - 7x + 10$$, so the polynomial becomes: $$x^3 - 8x^2 + 17x - 10 = (x - 1)(x^2 - 7x + 10)$$ ### Step 3: Factor the quadratic Next, we factor the quadratic $$x^2 - 7x + 10$$. We need two numbers that multiply to $$10$$ and add to $$-7$$. These numbers are $$-5$$ and $$-2$$. Thus: $$x^2 - 7x + 10 = (x - 5)(x - 2)$$ ### Step 4: Write the final factored form Now, combining everything: $$x^3 - 8x^2 + 17x - 10 = (x - 1)(x - 5)(x - 2)$$ This is the fully factored form of the polynomial. Would you like more details on any step or have any other questions? Here are 5 related questions: 1. How does the Factor Theorem help in finding the factors of polynomials? 2. What are the other methods for factoring a cubic polynomial? 3. Can synthetic division be used with non-integer roots? 4. How do you check if a given factorization is correct? 5. What happens if a cubic polynomial has complex roots? **Tip**: Always check for simple integer roots first, as they can significantly speed up the factoring process.