Which of the following is equivalent to $\left(p^3\right)\left(2 p^2-4 p\right)\left(3 p^2-1\right) ?$
A. $\left(p^3\right)\left(6 p^4-12 p^3-2 p^2+4 p\right)$
B. $\left(p^3\right)\left(6 p^4+4 p\right)$
C. $\left(2 p^6-4 p^3\right)\left(3 p^2-1\right)$
D. $\left(2 p^5-4 p^4\right)\left(3 p^5-p^3\right)$
Answer (1)
Expand the product of the polynomials: ( 2 p 2 − 4 p ) ( 3 p 2 − 1 ) = 6 p 4 − 2 p 2 − 12 p 3 + 4 p .
Multiply the result by p 3 : p 3 ( 6 p 4 − 12 p 3 − 2 p 2 + 4 p ) = 6 p 7 − 12 p 6 − 2 p 5 + 4 p 4 .
Compare the simplified expression with the given options.
The equivalent expression is ( p 3 ) ( 6 p 4 − 12 p 3 − 2 p 2 + 4 p ) .
Explanation
Understanding the Problem We are given the expression ( p 3 ) ( 2 p 2 − 4 p ) ( 3 p 2 − 1 ) and four other expressions. Our goal is to find which of the four expressions is equivalent to the given expression.
Expanding the terms First, we need to expand the terms ( 2 p 2 − 4 p ) ( 3 p 2 − 1 ) .
Calculating the product Expanding ( 2 p 2 − 4 p ) ( 3 p 2 − 1 ) gives us: ( 2 p 2 ) ( 3 p 2 ) + ( 2 p 2 ) ( − 1 ) + ( − 4 p ) ( 3 p 2 ) + ( − 4 p ) ( − 1 ) = 6 p 4 − 2 p 2 − 12 p 3 + 4 p
Multiplying by p^3 Next, we multiply the result by p 3 :
p 3 ( 6 p 4 − 2 p 2 − 12 p 3 + 4 p ) = 6 p 7 − 2 p 5 − 12 p 6 + 4 p 4
Rearranging terms Rearranging the terms, we get: 6 p 7 − 12 p 6 − 2 p 5 + 4 p 4
Checking the options Now we check the given options to see which one matches our result: Option 1: ( p 3 ) ( 6 p 4 − 12 p 3 − 2 p 2 + 4 p ) = 6 p 7 − 12 p 6 − 2 p 5 + 4 p 4 . This matches our result. Option 2: ( p 3 ) ( 6 p 4 + 4 p ) = 6 p 7 + 4 p 4 . This does not match. Option 3: ( 2 p 6 − 4 p 3 ) ( 3 p 2 − 1 ) = 6 p 8 − 2 p 6 − 12 p 5 + 4 p 3 . This does not match. Option 4: ( 2 p 5 − 4 p 4 ) ( 3 p 5 − p 3 ) = 6 p 10 − 2 p 8 − 12 p 9 + 4 p 7 . This does not match.
Final Answer Therefore, the equivalent expression is ( p 3 ) ( 6 p 4 − 12 p 3 − 2 p 2 + 4 p ) .
Examples
Polynomials are used to model curves and shapes in various fields, such as engineering, computer graphics, and economics. For instance, engineers use polynomials to design bridges and buildings, ensuring structural integrity and stability. In computer graphics, polynomials are used to create smooth curves and surfaces for 3D models and animations. Economists use polynomials to model economic growth and predict future trends. Understanding how to manipulate and simplify polynomial expressions is crucial for solving real-world problems in these fields.
