Fill in the missing pieces in the factorization of x3+2x2−9x−18
.
(x2− )(x+ )
The missing pieces in the factorization of x3+2x2−9x−18 are (x+3)(x-2).
To fill in the missing pieces in the factorization of x^3+2x^2-9x-18, we need to find two binomials that, when multiplied, result in the given expression.
To start, we can look at the coefficient of the highest power term, which is 1. This means that one of the terms in each binomial will have an x in it.
Next, we need to find the factors of the constant term, which is -18. The factors of -18 are ±1, ±2, ±3, ±6, ±9, and ±18. Since the coefficient of the x term in the original expression is -9, we want the sum of the two factors to be -9.
By trying some of the factors, we can see that -3 and 6 are the factors that add up to -9. Therefore, we can replace the missing pieces as follows:
(x^2-3x)(x+6)
So, the factorization of x^3+2x^2-9x-18 is (x^2-3x)(x+6).
To fill in the missing pieces in the factorization of the expression x^3 + 2x^2 - 9x - 18, we need to find two binomials (a polynomial with two terms) whose product is equal to the given expression.
1. First, we can attempt to find any possible rational roots using the Rational Root Theorem. The rational root theorem states that if a polynomial has a rational root (p/q), where p is a factor of the constant term (-18 in this case), and q is a factor of the leading coefficient (1 in this case), then the polynomial can be factored using (x-p/q) as a factor.
The factors of -18 are ±1, ±2, ±3, ±6, ±9, ±18, and the factors of 1 are ±1. So, the possible rational roots are ±1, ±2, ±3, ±6, ±9, and ±18.
2. Next, we can try substituting each of the possible rational roots into the expression and see if any of them make the expression equal to zero.
Let's try with x = 1:
(1)^3 + 2(1)^2 - 9(1) - 18 = 1 + 2 - 9 - 18 = -24
x = 1 is not a root.
Let's try with x = -1:
(-1)^3 + 2(-1)^2 - 9(-1) - 18 = -1 + 2 + 9 - 18 = -8
x = -1 is not a root.
Let's try with x = 2:
(2)^3 + 2(2)^2 - 9(2) - 18 = 8 + 8 - 18 - 18 = -20
x = 2 is not a root.
Let's try with x = -2:
(-2)^3 + 2(-2)^2 - 9(-2) - 18 = -8 + 8 + 18 - 18 = 0
x = -2 is a root.
3. Once we find a root, we can factor the given expression using long division or synthetic division, dividing by (x - root). In this case, we divide x^3 + 2x^2 - 9x - 18 by (x + 2).
Performing the division, we get:
(x^3 + 2x^2 - 9x - 18) / (x + 2) = x^2 - 9
So, we have factored x^3 + 2x^2 - 9x - 18 as:
(x + 2)(x^2 - 9)
Now, we can factor the quadratic term further:
(x + 2)(x + 3)(x - 3)
Therefore, the missing pieces in the factorization are (x + 3)(x - 3). The complete factorization is:
(x + 2)(x + 3)(x - 3).