$(2 p^2+7 p-3)(8 p-4)$
A) $40 p^3+24 p^2+32 p+48$
B) $15 p^3+16 p^2-24 p-64$
C) $16 p^3+48 p^2-52 p+12$
D) $12 p^3+34 p^2-18 p-16$
E) $40 p^3+6 p^2+4 p-2$
Answer (2)
Expand the expression ( 2 p 2 + 7 p − 3 ) ( 8 p − 4 ) by multiplying each term in the first polynomial by each term in the second polynomial.
Distribute and simplify: 2 p 2 ( 8 p − 4 ) + 7 p ( 8 p − 4 ) − 3 ( 8 p − 4 ) = ( 16 p 3 − 8 p 2 ) + ( 56 p 2 − 28 p ) + ( − 24 p + 12 ) .
Combine like terms: 16 p 3 + ( − 8 p 2 + 56 p 2 ) + ( − 28 p − 24 p ) + 12 = 16 p 3 + 48 p 2 − 52 p + 12 .
The correct answer is 16 p 3 + 48 p 2 − 52 p + 12 , which corresponds to option C. 16 p 3 + 48 p 2 − 52 p + 12
Explanation
Understanding the Problem We are given the expression ( 2 p 2 + 7 p − 3 ) ( 8 p − 4 ) and five possible simplified forms. Our goal is to expand the given expression and determine which of the five options matches the correct expansion.
Expanding the Expression To expand the expression ( 2 p 2 + 7 p − 3 ) ( 8 p − 4 ) , we multiply each term in the first polynomial by each term in the second polynomial:
( 2 p 2 + 7 p − 3 ) ( 8 p − 4 ) = 2 p 2 ( 8 p − 4 ) + 7 p ( 8 p − 4 ) − 3 ( 8 p − 4 )
Distributing the Terms Now, we distribute each term:
2 p 2 ( 8 p − 4 ) = 16 p 3 − 8 p 2 7 p ( 8 p − 4 ) = 56 p 2 − 28 p − 3 ( 8 p − 4 ) = − 24 p + 12
Combining Like Terms Next, we combine the results:
( 16 p 3 − 8 p 2 ) + ( 56 p 2 − 28 p ) + ( − 24 p + 12 ) = 16 p 3 + ( − 8 p 2 + 56 p 2 ) + ( − 28 p − 24 p ) + 12
Combining like terms, we get:
16 p 3 + 48 p 2 − 52 p + 12
Identifying the Matching Option Comparing the simplified expression 16 p 3 + 48 p 2 − 52 p + 12 to the five given options, we find that it matches option C.
Final Answer Therefore, the correct expansion of the given expression is 16 p 3 + 48 p 2 − 52 p + 12 .
Examples
Polynomial expansion is a fundamental concept in algebra and is used in various fields such as physics, engineering, and computer science. For example, when designing a bridge, engineers use polynomial equations to model the load distribution and stress on different parts of the structure. Expanding these polynomials helps them understand how the bridge will behave under different conditions and ensure its safety and stability. Similarly, in computer graphics, polynomial equations are used to create curves and surfaces, and expanding these equations allows developers to manipulate and render complex shapes.
The expansion of ( 2 p 2 + 7 p − 3 ) ( 8 p − 4 ) leads to the result of 16 p 3 + 48 p 2 − 52 p + 12 . The correct corresponding option from the choices given is option C. Therefore, the chosen answer is C.
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