Factor completely $-5 x^3+10 x^2$
A. Prime
B. $-x\left(5 x^2-10 x\right)$
C. $5 x\left(-x^2+2 x\right)$
D. $-5 x^2(x-2)$
Answer (2)
Identify the greatest common factor (GCF) of the terms: The GCF of − 5 x 3 and 10 x 2 is − 5 x 2 .
Factor out the GCF from the expression: − 5 x 3 + 10 x 2 = − 5 x 2 ( x − 2 ) .
Check if the remaining expression can be factored further: x − 2 cannot be factored further.
State the completely factored form: − 5 x 2 ( x − 2 ) .
Explanation
Understanding the Problem We are asked to factor the expression − 5 x 3 + 10 x 2 completely. This means we want to write the expression as a product of simpler expressions such that no further factoring is possible.
Finding the Greatest Common Factor First, we look for the greatest common factor (GCF) of the terms − 5 x 3 and 10 x 2 . The GCF of the coefficients − 5 and 10 is − 5 . The GCF of x 3 and x 2 is x 2 . Therefore, the GCF of − 5 x 3 and 10 x 2 is − 5 x 2 .
Factoring out the GCF Now, we factor out the GCF − 5 x 2 from the expression: − 5 x 3 + 10 x 2 = − 5 x 2 ( x ) + ( − 5 x 2 ) ( − 2 ) = − 5 x 2 ( x − 2 ) So, we have − 5 x 3 + 10 x 2 = − 5 x 2 ( x − 2 ) .
Final Factored Form The expression x − 2 cannot be factored further. Therefore, the completely factored form of the given expression is − 5 x 2 ( 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 complex equations when designing structures or circuits. In economics, factoring can help in analyzing cost and revenue functions to determine break-even points. Additionally, factoring is crucial in cryptography for securing data and communications.
The completely factored form of − 5 x 3 + 10 x 2 is − 5 x 2 ( x − 2 ) . This is achieved by factoring out the greatest common factor, which is − 5 x 2 . Thus, the correct answer is option D: − 5 x 2 ( x − 2 ) .
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