Simplify.
$-2 x y z(x y+3 y z)$
A. $-2 x^2 y^2 z-6 x y^2 z^2$
B. $-2 x^2 y^2 z+3 y^2$
C. $-2 x^2 y^2+3 y^2$
D. $-2 x^2 y^2 z+6 x y^2 z^2$
Answer (2)
Distribute − 2 x yz to both terms inside the parentheses: − 2 x yz ( x y + 3 yz ) = − 2 x yz ( x y ) + ( − 2 x yz ) ( 3 yz ) .
Multiply the terms: − 2 x yz ( x y ) = − 2 x 2 y 2 z and ( − 2 x yz ) ( 3 yz ) = − 6 x y 2 z 2 .
Combine the terms: − 2 x 2 y 2 z − 6 x y 2 z 2 .
The simplified expression is − 2 x 2 y 2 z − 6 x y 2 z 2 .
Explanation
Understanding the problem We are given the expression − 2 x yz ( x y + 3 yz ) and asked to simplify it. This involves distributing the term − 2 x yz to both terms inside the parentheses.
Applying the distributive property We apply the distributive property: − 2 x yz ( x y + 3 yz ) = − 2 x yz ( x y ) + ( − 2 x yz ) ( 3 yz ) .
Multiplying the terms Now, we multiply the terms: − 2 x yz ( x y ) = − 2 x 2 y 2 z and ( − 2 x yz ) ( 3 yz ) = − 6 x y 2 z 2 .
Combining the terms Combining the terms, we get: − 2 x 2 y 2 z − 6 x y 2 z 2 .
Final Answer Comparing the simplified expression with the given options, we see that it matches option A. Therefore, the simplified expression is − 2 x 2 y 2 z − 6 x y 2 z 2 .
Examples
Understanding how to simplify algebraic expressions like this is crucial in many areas, such as physics and engineering, where you often need to manipulate equations to solve for unknown variables. For example, if you're calculating the volume of a complex shape, you might need to simplify an expression involving variables representing length, width, and height. Simplifying expressions allows you to work with manageable equations and find accurate solutions. This skill is also fundamental in computer science when optimizing code or designing algorithms.
The simplified expression for − 2 x yz ( x y + 3 yz ) is − 2 x 2 y 2 z − 6 x y 2 z 2 , which corresponds to option A. This involves applying the distributive property to multiply and combine like terms. Thus, the correct answer is option A.
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