Factor completely $3 x^3+6 x^2-24 x$
A. $3 x(x-4)(x+2)$
B. $3 x(x+4)(x-2)$
C. $3 x(x-3)(x+2)$
D. $3 x(x+3)(x+2)$
Answer (1)
Factor out the greatest common factor: 3 x 3 + 6 x 2 − 24 x = 3 x ( x 2 + 2 x − 8 ) .
Factor the quadratic expression: x 2 + 2 x − 8 = ( x + 4 ) ( x − 2 ) .
Combine the factors: 3 x ( x + 4 ) ( x − 2 ) .
The completely factored form is: 3 x ( x + 4 ) ( x − 2 ) .
Explanation
Understanding the Problem We are given the polynomial 3 x 3 + 6 x 2 − 24 x and asked to factor it completely.
Factoring out the GCF First, we look for the greatest common factor (GCF) of all the terms. The GCF of 3 x 3 , 6 x 2 , and − 24 x is 3 x . Factoring out 3 x from the polynomial, we get: 3 x 3 + 6 x 2 − 24 x = 3 x ( x 2 + 2 x − 8 )
Factoring the Quadratic Expression Now, we need to factor the quadratic expression x 2 + 2 x − 8 . We are looking for two numbers that multiply to -8 and add to 2. These numbers are 4 and -2. Therefore, we can write the quadratic expression as: x 2 + 2 x − 8 = ( x + 4 ) ( x − 2 )
Final Factored Form Finally, we substitute the factored quadratic expression back into the expression we obtained after factoring out the GCF: 3 x ( x 2 + 2 x − 8 ) = 3 x ( x + 4 ) ( x − 2 ) Thus, the completely factored form of the polynomial is 3 x ( x + 4 ) ( x − 2 ) .
Final Answer The completely factored form of the polynomial 3 x 3 + 6 x 2 − 24 x is 3 x ( x + 4 ) ( x − 2 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify equations when designing structures or analyzing circuits. In economics, factoring can be used to model and predict market behavior. Factoring also plays a crucial role in cryptography, where it is used to create and break codes. Understanding how to factor polynomials allows us to solve complex problems in various fields.
