Which of the following is the factored form of x³ + 3x² - 10x - 24?
A. (x + 4)(x - 3)(x + 2)
B. (x - 4)(x - 3)(x - 2)
C. (x - 4)(x - 3)(x + 2)
D. (x + 4)(x + 3)(x - 2)
Try dividing by x+2 and you will see that you have
(x+2)(x^2+x-12)
Now repeat to see whether the answer is A or C
To factorize the expression x³ + 3x² - 10x - 24, we can use the grouping method or synthetic division. Here's the step-by-step process for factoring the given expression:
Step 1: Group the terms:
(x³ + 3x²) + (-10x - 24)
Step 2: Factor out the common terms from each group:
x²(x + 3) - 2(5x + 12)
Step 3: Factor out the common factor from both terms:
x²(x + 3) - 2(5x + 12)
Step 4: Factor out the common factor of (x + 3) from the first group:
x²(x + 3) - 2(5x + 12)
Step 5: Apply the distributive property:
(x + 3)(x² - 2(5x + 12))
Step 6: Simplify the expression:
(x + 3)(x² - 10x - 24)
Therefore, the factored form of the expression x³ + 3x² - 10x - 24 is (x + 3)(x² - 10x - 24).
Comparing this with the given options, the correct answer is C. (x - 4)(x - 3)(x + 2).
To find the factored form of a polynomial, we need to factor it completely. In this case, we are given the polynomial x³ + 3x² - 10x - 24, and we need to determine which of the answer choices represents its factored form.
To factor the polynomial, we can start by looking for any common factors. In this case, there are no common factors among the terms.
Next, we can look for any apparent patterns or strategies that can help us factor the polynomial. One common approach is to try to identify any grouping or factoring patterns.
In this case, we can try factoring by grouping. Let's group the terms into pairs:
(x³ + 3x²) + (-10x - 24)
Now, let's factor out the greatest common factor from each pair separately:
x²(x + 3) - 2(5x + 12)
Now, we have two terms that can be factored further. Let's factor each of these separately:
x²(x + 3) - 2(5x + 12)
(x + 3)(x² - 2(5x + 12))
The term x² - 2(5x + 12) can be further factored as the difference of squares. We have -2(5x + 12), which can be written as -2 * 5 * (x + 3 * 2), or -10(x + 6). Therefore, we have:
(x + 3)(x² - 10(x + 6))
Now, let's distribute -10 in the second term:
(x + 3)(x² - 10x - 60)
Now, we have factored x³ + 3x² - 10x - 24 into (x + 3)(x² - 10x - 60).
Comparing this factored form with the answer choices, we can see that the correct answer is option C. Therefore, the factored form of x³ + 3x² - 10x - 24 is (x - 4)(x - 3)(x + 2).