Fill in the missing pieces in the factorization of x3+2x2−9x−18
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The factorization of x^3 + 2x^2 - 9x - 18 is:
(x + 3)(x - 2)(x + 3)
To factorize the expression x^3 + 2x^2 - 9x - 18, we can use the grouping method. Here are the steps:
Step 1: Group the terms in pairs:
(x^3 + 2x^2) - (9x + 18)
Step 2: Factor out the greatest common factor from each pair:
x^2(x + 2) - 9(x + 2)
Step 3: Now we can see that (x + 2) is a common factor. Factor it out:
(x + 2)(x^2 - 9)
Step 4: The expression x^2 - 9 is a difference of squares, which can be further factorized as:
(x + 2)(x + 3)(x - 3)
So the complete factorization of x^3 + 2x^2 - 9x - 18 is (x + 2)(x + 3)(x - 3).
To fill in the missing pieces in the factorization of x^3 + 2x^2 - 9x - 18, we need to first identify if there are any common factors among the terms.
Step 1: Check for common factors
Looking at the expression, we can see that there are no explicit common factors among all the terms.
Step 2: Test possible linear factors
Since the degree of the polynomial is 3, the possible linear factors can be found by considering the factors of the constant term (18 in this case, with positive and negative signs) divided by the factors of the leading coefficient (1 in this case, since it is a monic polynomial).
The possible linear factors are: ±1, ±2, ±3, ±6, ±9, ±18.
We can test these factors by substituting them into the polynomial to see if any of them yield a remainder of zero when evaluated. By the Remainder Theorem, if a factor is valid, it means that the polynomial can be evenly divided by the linear factor.
Using synthetic division, we can test each of these linear factors one by one to see which ones result in a remainder of zero.
Step 3: Synthetic division
We will use synthetic division to test these factors. Let's start with the factor x - 1.
1 | 1 2 -9 -18
| 1 3 -6
-------------------
1 3 -6 -24
Here, the remainder is -24, so x - 1 is not a factor.
Next, let's try x + 1.
-1 | 1 2 -9 -18
| -1 -1 10
--------------------
1 1 -10 -8
The remainder is -8, so x + 1 is not a factor.
Trying x - 2:
2 | 1 2 -9 -18
| 2 8 -2
--------------------
1 4 -1 -20
The remainder is -20, so x - 2 is not a factor.
Next, let's try x + 2.
-2 | 1 2 -9 -18
| -2 0 18
--------------------
1 0 9 0
The remainder is 0, so x + 2 is a factor.
Step 4: Factoring the polynomial
Since we found that x + 2 is a factor, we can now divide the polynomial by x + 2 using synthetic division to obtain the remaining quadratic expression:
-2 | 1 0 9
| -2 4
------------------
1 -2 13
The quotient is x^2 - 2x + 13.
Therefore, the factorization of x^3 + 2x^2 - 9x - 18 is (x + 2)(x^2 - 2x + 13).