Factor completely x3 − 2x2 − 5x + 10.

To factor the polynomial \( x^3 - 2x^2 - 5x + 10 \) completely, we follow these steps:

1. **Find the roots of the polynomial** using the Rational Root Theorem. According to this theorem, the possible rational roots of the polynomial \( x^3 - 2x^2 - 5x + 10 \) are the factors of the constant term (10) divided by the factors of the leading coefficient (1). This gives us the potential roots: \(\pm 1, \pm 2, \pm 5, \pm 10\).

2. **Test the potential roots** by substituting them into the polynomial to see if they yield zero.

Let's start with \( x = 1 \):
\[ 1^3 - 2(1^2) - 5(1) + 10 = 1 - 2 - 5 + 10 = 4 \]
So, \( x = 1 \) is not a root.

Next, try \( x = -1 \):
\[ (-1)^3 - 2(-1)^2 - 5(-1) + 10 = -1 - 2 + 5 + 10 = 12 \]
So, \( x = -1 \) is not a root.

Next, try \( x = 2 \):
\[ 2^3 - 2(2^2) - 5(2) + 10 = 8 - 8 - 10 + 10 = 0 \]
So, \( x = 2 \) is a root.

Since \( x = 2 \) is a root, \( x - 2 \) is a factor of the polynomial. We can now perform polynomial division to factor out \( x - 2 \).

3. **Divide the polynomial** \( x^3 - 2x^2 - 5x + 10 \) by \( x - 2 \).

Using synthetic division:

\[
\begin{array}{r|rrrr}
2 & 1 & -2 & -5 & 10 \\
\hline
  &   & 2 & 0 & -10 \\
\hline
  & 1 & 0 & -5 & 0 \\
\end{array}
\]

The result of the division is \( x^2 - 5 \).

4. **Factor the quotient** \( x^2 - 5 \). This can be factored as a difference of squares:
\[ x^2 - 5 = (x - \sqrt{5})(x + \sqrt{5}) \]

5. **Combine all the factors**:
\[ x^3 - 2x^2 - 5x + 10 = (x - 2)(x - \sqrt{5})(x + \sqrt{5}) \]

So, the completely factored form of \( x^3 - 2x^2 - 5x + 10 \) is:
\[ (x - 2)(x - \sqrt{5})(x + \sqrt{5}) \]

Would you like any further details or have any questions?

Here are 8 related questions that you might find useful:
1. How do you use the Rational Root Theorem to find possible rational roots of a polynomial?
2. What are the steps involved in synthetic division?
3. How can you verify if a given number is a root of a polynomial?
4. What is the difference of squares and how do you factor it?
5. How do you factor higher degree polynomials?
6. Can every polynomial be factored completely into linear factors?
7. What are complex roots and how do they relate to polynomial factorization?
8. How do you find the roots of a polynomial equation using a graphing calculator?

**Tip:** When using the Rational Root Theorem, always list all possible rational roots to ensure you consider every potential candidate for factorization.

Learn how to factor completely the polynomial x^3 - 2x^2 - 5x + 10 with detailed step-by-step solutions. This problem involves polynomial factorization, synthetic division, and the application of the Rational Root Theorem. Suitable for students in Grades 10-12.

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