Which expression is the simplest form of $-(3x^3+x^2)+2(x^3-4x^2)$?
A. $5x^3-8x^2$
B. $-x^3-9x^2$
C. $5x^3-7x^2$
D. $-x^3-3x^2$
Answer (2)
Distribute the negative sign: − ( 3 x 3 + x 2 ) = − 3 x 3 − x 2 .
Distribute the 2: 2 ( x 3 − 4 x 2 ) = 2 x 3 − 8 x 2 .
Combine like terms: ( − 3 x 3 − x 2 ) + ( 2 x 3 − 8 x 2 ) = ( − 3 x 3 + 2 x 3 ) + ( − x 2 − 8 x 2 ) .
Simplify: ( − 3 + 2 ) x 3 + ( − 1 − 8 ) x 2 = − x 3 − 9 x 2 . The answer is − x 3 − 9 x 2 .
Explanation
Understanding the Problem We are asked to simplify the expression − ( 3 x 3 + x 2 ) + 2 ( x 3 − 4 x 2 ) . This involves distributing the negative sign and the 2, and then combining like terms.
Distributing the Negative Sign First, distribute the negative sign in the first term: − ( 3 x 3 + x 2 ) = − 3 x 3 − x 2
Distributing the 2 Next, distribute the 2 in the second term: 2 ( x 3 − 4 x 2 ) = 2 x 3 − 8 x 2
Combining Like Terms Now, combine the like terms: ( − 3 x 3 − x 2 ) + ( 2 x 3 − 8 x 2 ) = ( − 3 x 3 + 2 x 3 ) + ( − x 2 − 8 x 2 )
Simplifying the Expression Simplify the expression by adding the coefficients of the x 3 terms and the x 2 terms: ( − 3 + 2 ) x 3 + ( − 1 − 8 ) x 2 = − 1 x 3 + ( − 9 ) x 2 = − x 3 − 9 x 2
Final Answer The simplified expression is − x 3 − 9 x 2 . Therefore, the correct answer is B.
Examples
Simplifying polynomial expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use polynomials to model the behavior of structures under stress. Simplifying these polynomial expressions allows them to more easily analyze the structural integrity and make design decisions. Similarly, economists use polynomials to model cost and revenue functions, and simplifying these expressions helps them to optimize production and pricing strategies.
The simplest form of the expression − ( 3 x 3 + x 2 ) + 2 ( x 3 − 4 x 2 ) is − x 3 − 9 x 2 . Therefore, the correct answer is B. This involves distributing terms and combining like terms carefully.
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