What is the product of [tex]$\left(p^3\right)\left(2 p^2-4 p\right)\left(3 p^2-1\right)$[/tex]?
A. [tex]$6 p^7+4 p^4$[/tex]
B. [tex]$6 p^7-2 p^5-12 p^5+4 p^3$[/tex]
C. [tex]$6 p^7-12 p^6-2 p^5+4 p^4$[/tex]
D. [tex]$6 p^{12}-14 p^6+4 p^3$[/tex]
Answer (1)
Multiply ( p 3 ) by ( 2 p 2 − 4 p ) to get 2 p 5 − 4 p 4 .
Multiply the result ( 2 p 5 − 4 p 4 ) by ( 3 p 2 − 1 ) to get 6 p 7 − 2 p 5 − 12 p 6 + 4 p 4 .
Rearrange the terms in decreasing order of the exponent of p .
The final product is 6 p 7 − 12 p 6 − 2 p 5 + 4 p 4 .
Explanation
Understanding the Problem We are asked to find the product of three expressions: ( p 3 ) , ( 2 p 2 − 4 p ) , and ( 3 p 2 − 1 ) . Our goal is to multiply these expressions together and simplify the result.
Multiplying the First Two Expressions First, let's multiply ( p 3 ) and ( 2 p 2 − 4 p ) .
p 3 ( 2 p 2 − 4 p ) = p 3 ( 2 p 2 ) − p 3 ( 4 p ) = 2 p 5 − 4 p 4
Multiplying by the Third Expression Now, let's multiply the result ( 2 p 5 − 4 p 4 ) by the third expression ( 3 p 2 − 1 ) .
( 2 p 5 − 4 p 4 ) ( 3 p 2 − 1 ) = 2 p 5 ( 3 p 2 − 1 ) − 4 p 4 ( 3 p 2 − 1 ) = 2 p 5 ( 3 p 2 ) − 2 p 5 ( 1 ) − 4 p 4 ( 3 p 2 ) + 4 p 4 ( 1 ) = 6 p 7 − 2 p 5 − 12 p 6 + 4 p 4
Rearranging the Terms Finally, let's rearrange the terms in decreasing order of the exponent of p .
6 p 7 − 12 p 6 − 2 p 5 + 4 p 4
Final Answer The product of the given expressions is 6 p 7 − 12 p 6 − 2 p 5 + 4 p 4 .
Examples
Understanding polynomial multiplication is crucial in various fields, such as engineering and computer science. For instance, when designing filters for signal processing, engineers often need to multiply polynomials to determine the overall transfer function of the filter. Similarly, in computer graphics, polynomial multiplication is used in curve and surface modeling to create smooth and realistic shapes. This algebraic skill is fundamental for solving complex problems in these areas.
