$(x^2+x-2)(4x^2-8x)
Answer (2)
Factor the quadratic expression: x 2 + x − 2 = ( x + 2 ) ( x − 1 ) .
Factor the second expression: 4 x 2 − 8 x = 4 x ( x − 2 ) .
Multiply the factored expressions: ( x + 2 ) ( x − 1 ) ( 4 x ) ( x − 2 ) .
The simplified expression is 4 x ( x − 1 ) ( x − 2 ) ( x + 2 ) .
Explanation
Understanding the Problem We are given the expression ( x 2 + x − 2 ) ( 4 x 2 − 8 x ) and our goal is to simplify it. To do this, we will first factor each part of the expression and then multiply them together.
Factoring the Quadratic Expression First, let's factor the quadratic expression x 2 + x − 2 . We are looking for two numbers that multiply to -2 and add to 1. These numbers are 2 and -1. So, we can factor the quadratic as follows: x 2 + x − 2 = ( x + 2 ) ( x − 1 )
Factoring the Second Expression Next, let's factor the expression 4 x 2 − 8 x . We can factor out a 4 x from both terms: 4 x 2 − 8 x = 4 x ( x − 2 )
Multiplying the Factored Expressions Now, we multiply the factored expressions together: ( x 2 + x − 2 ) ( 4 x 2 − 8 x ) = ( x + 2 ) ( x − 1 ) ( 4 x ) ( x − 2 ) We can rearrange the terms to make it look nicer: = 4 x ( x − 1 ) ( x − 2 ) ( x + 2 )
Final Answer So, the simplified expression is 4 x ( x − 1 ) ( x − 2 ) ( x + 2 ) .
Examples
Simplifying polynomial expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use polynomial expressions to model the behavior of physical systems, such as the trajectory of a projectile or the stress on a bridge. By simplifying these expressions, engineers can more easily analyze and predict the behavior of these systems. Another example is in computer graphics, where polynomial expressions are used to create smooth curves and surfaces. Simplifying these expressions can improve the efficiency of the rendering process.
To simplify the expression ( x 2 + x − 2 ) ( 4 x 2 − 8 x ) , we factor each part as follows: x 2 + x − 2 = ( x + 2 ) ( x − 1 ) and 4 x 2 − 8 x = 4 x ( x − 2 ) . Multiplying these together gives us the final result of 4 x ( x − 1 ) ( x − 2 ) ( x + 2 ) .
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