Factor completely $x^3+3 x^2-4 x-12$
A. $(x+2)(x-2)(x+3)$
B. $(x+2)(x-2)(x+3)(x-3)$
C. $\left(x^2+4\right)(x+3)$
D. $\left(x^2-4\right)(x+3)$
Answer (1)
Group the terms: ( x 3 + 3 x 2 ) + ( − 4 x − 12 ) .
Factor out common factors: x 2 ( x + 3 ) − 4 ( x + 3 ) .
Factor out the common binomial: ( x + 3 ) ( x 2 − 4 ) .
Factor the difference of squares: ( x + 3 ) ( x − 2 ) ( x + 2 ) . The final factored form is ( x + 2 ) ( x − 2 ) ( x + 3 ) .
Explanation
Problem Analysis We are given the cubic polynomial x 3 + 3 x 2 − 4 x − 12 and asked to factor it completely.
Grouping Terms We can use factoring by grouping to factor this polynomial. First, we group the first two terms and the last two terms: ( x 3 + 3 x 2 ) + ( − 4 x − 12 ) .
Factoring out Common Factors Next, we factor out the greatest common factor from each group. From the first group, x 3 + 3 x 2 , we can factor out x 2 , which gives us x 2 ( x + 3 ) . From the second group, − 4 x − 12 , we can factor out − 4 , which gives us − 4 ( x + 3 ) . So we have x 2 ( x + 3 ) − 4 ( x + 3 ) .
Factoring out the Binomial Now, we can factor out the common binomial factor ( x + 3 ) from the entire expression: ( x + 3 ) ( x 2 − 4 ) .
Factoring the Difference of Squares Finally, we recognize that x 2 − 4 is a difference of squares, which can be factored as ( x − 2 ) ( x + 2 ) . Therefore, the completely factored form of the polynomial is ( x + 3 ) ( x − 2 ) ( x + 2 ) .
Final Answer Thus, the completely factored form of the given polynomial is ( x + 2 ) ( x − 2 ) ( x + 3 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model economic growth, and computer scientists use it to design algorithms. Imagine you are designing a bridge and need to ensure it can withstand certain loads. By expressing the load as a polynomial and factoring it, you can identify critical points where the load is maximized or minimized, allowing you to design a safer and more efficient structure. Factoring helps simplify complex equations, making them easier to analyze and solve.
