Fill in the missing pieces in the factorization of x3+2x2−9x−18
.
(1 point)
(x2−
)(x+
)
(x2−3x)(x+6)
To factorize the expression x^3 + 2x^2 - 9x - 18, we need to find two binomials whose product gives us this expression.
Step 1: Look for possible common factors.
Because there are no common factors among all the terms, we move to the next step.
Step 2: Look for possible factor pairs for the constant term (-18).
The factor pairs of -18 are: (-1, 18), (1, -18), (-2, 9), (2, -9), (-3, 6), (3, -6), (-6, 3), and (6, -3).
Step 3: Test the factor pairs.
To test each factor pair, we need to check if substituting the factors into the expression results in zero. Let's try the factor pair (-2, 9) first.
Substituting -2 and 9 into the expression:
(-2)^3 + 2(-2)^2 - 9(-2) - 18 = -8 + 8 + 18 - 18 = 0
Since it results in zero, this means that x + 2 is one of the factors of the polynomial.
Step 4: Divide the polynomial by the factor.
Using polynomial division, divide x^3 + 2x^2 - 9x - 18 by x + 2 to find the remaining factor.
x^2 - 7x + 9
x + 2 ) x^3 + 2x^2 - 9x - 18
- (x^3 + 2x^2)
__________
-11x - 18
-(-11x - 22)
__________
4
The result of the division is x^2 - 7x + 9. Therefore, the complete factorization of the expression x^3 + 2x^2 - 9x - 18 is (x + 2)(x^2 - 7x + 9).
To factorize the given expression x^3 + 2x^2 - 9x - 18, we need to find two binomials whose product is equal to the given expression.
Step 1: Look for the common factors, if any.
In this case, there are no common factors among the given terms.
Step 2: Start by finding the factors of the constant term (-18).
The factors of -18 are: ±1, ±2, ±3, ±6, ±9, ±18.
Step 3: Determine the possible rational roots based on the factors found.
The possible rational roots (also known as the rational zero theorem) can be determined by taking the factors of the constant term and dividing them by the factors of the leading coefficient (which is 1 in this case).
Possible rational roots = ±1, ±2, ±3, ±6, ±9, ±18
Step 4: Use the possible rational roots to perform synthetic division.
Using synthetic division or long division, you can try out each possible root to determine if it is a factor or not. By testing these roots, we can find two factors, one quadratic and one linear.
By synthetic division:
Trying x = -1:
-1 | 1 2 -9 -18
|________
-1 -1 10
|________ _______
1 1 -8
Since the remainder is 0, it means x = -1 is a factor.
Step 5: Rewrite the given expression using the factor obtained from synthetic division.
From the synthetic division, we found that x + 1 is a factor. So we can write:
x^3 + 2x^2 - 9x - 18 = (x + 1)(x^2 + x - 8)
Now we have the factorization:
(x + 1)(x^2 + x - 8)
So, the missing pieces in the factorization are (x + 1)(x^2 + x - 8).