$(x^2+x-2)(4x^2-8x)$
Answer (1)
Factor x 2 + x − 2 into ( x + 2 ) ( x − 1 ) .
Factor 4 x 2 − 8 x into 4 x ( x − 2 ) .
Multiply the factored expressions: ( x + 2 ) ( x − 1 ) ( 4 x ) ( x − 2 ) .
The simplified expression is 4 x ( x + 2 ) ( x − 1 ) ( 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. This involves factoring both quadratic expressions and then multiplying them together.
Factoring the First Quadratic 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. Therefore, we can factor the quadratic as ( 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, which gives us 4 x ( x − 2 ) .
Multiplying the Factored Expressions Now, we multiply the factored expressions together: ( x + 2 ) ( x − 1 ) ( 4 x ) ( x − 2 ) . We can rewrite this as 4 x ( x + 2 ) ( x − 1 ) ( x − 2 ) .
Final Simplified Expression So the simplified expression is 4 x ( x + 2 ) ( x − 1 ) ( x − 2 ) .
Examples
Understanding how to simplify polynomial expressions is crucial in many areas, such as physics and engineering, where complex equations need to be solved efficiently. For example, when analyzing the trajectory of a projectile, simplifying the equation of motion allows for easier calculation of the projectile's range and maximum height. Similarly, in circuit analysis, simplifying the equations describing the circuit's behavior helps in predicting its performance and designing it effectively. This skill is also fundamental in computer graphics for rendering complex scenes and animations.
