Questionfactor $x^{3}-2 x^{2}-x+2$

Studdy Solution

STEP 1

1. We are given a cubic polynomial $x^{3}-2x^{2}-x+2$ .
2. The goal is to factor this polynomial into simpler polynomials.
3. We will use methods such as factoring by grouping or the Rational Root Theorem to find factors.

STEP 2

1. Use the Rational Root Theorem to find possible rational roots.
2. Test the possible roots to find an actual root.
3. Use the found root to factor the polynomial.
4. Factor the resulting quadratic polynomial, if possible.

STEP 3

Apply the Rational Root Theorem to find possible rational roots. The possible rational roots are the factors of the constant term, $2$ , divided by the factors of the leading coefficient, $1$ .
Possible rational roots are: $\pm 1, \pm 2$ .

STEP 4

Test the possible roots by substituting them into the polynomial to see if they yield zero.
Test $x = 1$ :
$f(1) = 1^3 - 2 \times 1^2 - 1 + 2 = 1 - 2 - 1 + 2 = 0$
Since $f(1) = 0$ , $x = 1$ is a root.

STEP 5

Use the root $x = 1$ to factor the polynomial. Since $x = 1$ is a root, $(x - 1)$ is a factor. Perform polynomial division of $x^3 - 2x^2 - x + 2$ by $x - 1$ .
Using synthetic division:
$\begin{array}{r|rrrr} 1 & 1 & -2 & -1 & 2 \\ & & 1 & -1 & -2 \\ \hline & 1 & -1 & -2 & 0 \\ \end{array}$
The quotient is $x^2 - x - 2$ .

STEP 6

Factor the quadratic $x^2 - x - 2$ .
Find two numbers that multiply to $-2$ and add to $-1$ . These numbers are $-2$ and $1$ .
Thus, $x^2 - x - 2 = (x - 2)(x + 1)$ .

STEP 7

Combine the factors to write the complete factorization of the original polynomial:
$x^3 - 2x^2 - x + 2 = (x - 1)(x - 2)(x + 1)$

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