Factor the polynomial as the product of three binomials.
Note: Try using factor theorem for guessing some of the factors.
\[x^3 -8x^2 +17x -10=\]

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:

\[
\begin{array}{r|rrrr}
1 & 1 & -8 & 17 & -10 \\
  &   & 1  & -7 & 10 \\
\hline
  & 1 & -7 & 10 & 0
\end{array}
\]

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.

Factor the polynomial as the product of three binomials. Note: Try using factor theorem for guessing some of the factors.
x^3 - 8x^2 + 17x - 10 =

Learn how to factor the cubic polynomial x^3 - 8x^2 + 17x - 10 using the Factor Theorem. This detailed step-by-step guide covers synthetic division, quadratic factorization, and key algebraic concepts, making it suitable for Grades 10-12.

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