Question

decompose x^3-2x^2-5x+6

Answer 1

To decompose the polynomial \( x^3 - 2x^2 - 5x + 6 \), we can start by attempting to factor it. First, we can use the Rational Root Theorem to test for potential rational roots among the factors of the constant term, \( 6 \). The potential rational roots are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Let's test these values one by one:

1. For \( x = 1 \):
\[
1^3 - 2(1^2) - 5(1) + 6 = 1 - 2 - 5 + 6 = 0
\]
Thus, \( x = 1 \) is a root.

Now, we can perform polynomial long division, dividing \( x^3 - 2x^2 - 5x + 6 \) by \( x - 1 \).

**Long Division Steps:**
1. Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( (x - 1) \) to get \( x^3 - x^2 \).
3. Subtract: \( (x^3 - 2x^2 - 5x + 6) - (x^3 - x^2) = -x^2 - 5x + 6 \).
4. Divide \( -x^2 \) by \( x \) to get \( -x \).
5. Multiply \( -x \) by \( (x - 1) \) to get \( -x^2 + x \).
6. Subtract: \( (-x^2 - 5x + 6) - (-x^2 + x) = -6x + 6 \).
7. Divide \( -6x \) by \( x \) to get \( -6 \).
8. Multiply \( -6 \) by \( (x - 1) \) to get \( -6x + 6 \).
9. Subtract: \( (-6x + 6) - (-6x + 6) = 0 \).

So, we find the quotient is:
\[
x^2 - x - 6
\]

Next, we can factor \( x^2 - x - 6 \):
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

Thus, the factorization of the original polynomial is:
\[
x^3 - 2x^2 - 5x + 6 = (x - 1)(x - 3)(x + 2)
\]

In conclusion, the full factorization is:
\[
\boxed{(x - 1)(x - 3)(x + 2)}
\]