What is the product of $(2 p+7)\left(3 p^2+4 p-3\right)$?
A. $6 p^3+29 p^2-34 p+21$
B. $6 p^3+29 p^2-22 p+21$
C. $6 p^3+29 p^2+22 p-21$
D. $6 p^3+29 p^2+34 p-21
Answer (1)
Distribute the terms: Multiply each term in ( 2 p + 7 ) by each term in ( 3 p 2 + 4 p − 3 ) .
Expand the expression: 2 p ( 3 p 2 + 4 p − 3 ) + 7 ( 3 p 2 + 4 p − 3 ) = 6 p 3 + 8 p 2 − 6 p + 21 p 2 + 28 p − 21 .
Combine like terms: 6 p 3 + ( 8 p 2 + 21 p 2 ) + ( − 6 p + 28 p ) − 21 .
Simplify: The product is 6 p 3 + 29 p 2 + 22 p − 21 .
Explanation
Understanding the Problem We are given the expression ( 2 p + 7 ) ( 3 p 2 + 4 p − 3 ) and asked to find the product.
Applying the Distributive Property To find the product, we will use the distributive property to multiply each term in the first polynomial by each term in the second polynomial:
( 2 p + 7 ) ( 3 p 2 + 4 p − 3 ) = 2 p ( 3 p 2 + 4 p − 3 ) + 7 ( 3 p 2 + 4 p − 3 )
Expanding the Terms Now, we expand each term:
2 p ( 3 p 2 + 4 p − 3 ) = 6 p 3 + 8 p 2 − 6 p
7 ( 3 p 2 + 4 p − 3 ) = 21 p 2 + 28 p − 21
Combining the Expressions Next, we add the two expanded expressions:
( 6 p 3 + 8 p 2 − 6 p ) + ( 21 p 2 + 28 p − 21 ) = 6 p 3 + ( 8 p 2 + 21 p 2 ) + ( − 6 p + 28 p ) − 21
Simplifying the Expression Finally, we combine like terms to simplify the expression:
6 p 3 + ( 8 p 2 + 21 p 2 ) + ( − 6 p + 28 p ) − 21 = 6 p 3 + 29 p 2 + 22 p − 21
Final Answer Therefore, the product of ( 2 p + 7 ) ( 3 p 2 + 4 p − 3 ) is 6 p 3 + 29 p 2 + 22 p − 21 .
Examples
Polynomial multiplication is a fundamental concept in algebra and has numerous applications in real-world scenarios. For instance, engineers use polynomial multiplication to model and analyze systems, such as electrical circuits or mechanical structures. By representing system parameters as polynomials, engineers can predict the behavior of the system under different conditions. Similarly, economists use polynomial multiplication to model economic growth and predict future trends. Understanding polynomial multiplication is crucial for solving complex problems in various fields.
