To factorize the expression x^3 - 2x^2 - x - 2, we can use various techniques such as grouping or identifying common factors. In this case, we'll use the method of synthetic division to find one root of the polynomial, and then factor it accordingly.
Step 1: Find one root of the polynomial (let's call it "a").
By using trial and error, we can check the possible integer roots of the polynomial. We find that x = 2 is a root.
Step 2: Perform synthetic division to factorize the polynomial.
Now, divide x^3 - 2x^2 - x - 2 by (x - 2) using synthetic division:
```
2 | 1 - 2 - 1 - 2
| 2 0 -2
|____________________
1 0 -1 -4
```
The result of the synthetic division is the quotient: 1x^2 + 0x - 1 and the remainder: -4.
Step 3: Write the polynomial in factored form.
The factored form of the polynomial can be obtained as follows:
x^3 - 2x^2 - x - 2 = (x - 2)(x^2 + 0x - 1)
Now, we can simplify further:
x^2 + 0x - 1 = x^2 - 1
Finally, the fully factored form of the polynomial is:
x^3 - 2x^2 - x - 2 = (x - 2)(x^2 - 1)
Note that x^2 - 1 can be further factored as a difference of squares:
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To factorize the expression x^3 - 2x^2 - x - 2, we can use various techniques such as grouping or identifying common factors. In this case, we'll use the method of synthetic division to find one root of the polynomial, and then factor it accordingly.
Step 1: Find one root of the polynomial (let's call it "a").
By using trial and error, we can check the possible integer roots of the polynomial. We find that x = 2 is a root.
Step 2: Perform synthetic division to factorize the polynomial.
Now, divide x^3 - 2x^2 - x - 2 by (x - 2) using synthetic division:
```
2 | 1 - 2 - 1 - 2
| 2 0 -2
|____________________
1 0 -1 -4
```
The result of the synthetic division is the quotient: 1x^2 + 0x - 1 and the remainder: -4.
Step 3: Write the polynomial in factored form.
The factored form of the polynomial can be obtained as follows:
x^3 - 2x^2 - x - 2 = (x - 2)(x^2 + 0x - 1)
Now, we can simplify further:
x^2 + 0x - 1 = x^2 - 1
Finally, the fully factored form of the polynomial is:
x^3 - 2x^2 - x - 2 = (x - 2)(x^2 - 1)
Note that x^2 - 1 can be further factored as a difference of squares:
x^2 - 1 = (x + 1)(x - 1)
So, the complete factorization is:
x^3 - 2x^2 - x - 2 = (x - 2)(x + 1)(x - 1)