$\left(n^2-3 n+1\right)(2 n+3) =$
Answer (1)
Expand the expression by multiplying each term in the first factor by each term in the second factor.
Distribute and combine the terms: n 2 ( 2 n + 3 ) − 3 n ( 2 n + 3 ) + 1 ( 2 n + 3 ) = 2 n 3 + 3 n 2 − 6 n 2 − 9 n + 2 n + 3 .
Combine like terms: 2 n 3 + ( 3 n 2 − 6 n 2 ) + ( − 9 n + 2 n ) + 3 = 2 n 3 − 3 n 2 − 7 n + 3 .
The polynomial in standard form is 2 n 3 − 3 n 2 − 7 n + 3 .
Explanation
Understanding the Problem We are given the expression ( n 2 − 3 n + 1 ) ( 2 n + 3 ) and asked to expand it and write the result as a polynomial in standard form. Standard form means that the terms are ordered by decreasing degree.
Expanding the Expression To expand the expression, we multiply each term in the first factor by each term in the second factor:
( n 2 − 3 n + 1 ) ( 2 n + 3 ) = n 2 ( 2 n + 3 ) − 3 n ( 2 n + 3 ) + 1 ( 2 n + 3 )
Distributing Now, we distribute each term:
n 2 ( 2 n + 3 ) = 2 n 3 + 3 n 2
− 3 n ( 2 n + 3 ) = − 6 n 2 − 9 n
1 ( 2 n + 3 ) = 2 n + 3
Combining Terms Next, we combine these results:
2 n 3 + 3 n 2 − 6 n 2 − 9 n + 2 n + 3
Simplifying Now, we combine like terms:
2 n 3 + ( 3 n 2 − 6 n 2 ) + ( − 9 n + 2 n ) + 3 = 2 n 3 − 3 n 2 − 7 n + 3
Final Answer The resulting polynomial in standard form is 2 n 3 − 3 n 2 − 7 n + 3 .
Examples
Polynomial multiplication is used in various fields, such as engineering, physics, and computer science. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution and structural integrity. Multiplying these polynomials helps them understand how different factors interact and ensure the bridge's safety and stability. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces for realistic rendering.
