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The product of $\left(n^2-6 n+3\right)$ and $-4 n$ is $\square$ When this product is multiplied by $-n$, the result is $\square$
Answer (1)
First, distribute − 4 n to the expression ( n 2 − 6 n + 3 ) , resulting in − 4 n 3 + 24 n 2 − 12 n .
Then, distribute − n to the expression − 4 n 3 + 24 n 2 − 12 n , resulting in 4 n 4 − 24 n 3 + 12 n 2 .
The product of ( n 2 − 6 n + 3 ) and − 4 n is − 4 n 3 + 24 n 2 − 12 n .
When this product is multiplied by − n , the result is 4 n 4 − 24 n 3 + 12 n 2 .
Explanation
Understanding the Problem We are given the expression ( n 2 − 6 n + 3 ) and we need to multiply it by − 4 n . Then, we need to multiply the result by − n again. The objective is to find the simplified expressions after each multiplication.
Multiplying by -4n First, we multiply ( n 2 − 6 n + 3 ) by − 4 n . We distribute − 4 n to each term inside the parentheses: − 4 n ( n 2 − 6 n + 3 ) = − 4 n ( n 2 ) − 4 n ( − 6 n ) − 4 n ( 3 ) = − 4 n 3 + 24 n 2 − 12 n So, the product of ( n 2 − 6 n + 3 ) and − 4 n is − 4 n 3 + 24 n 2 − 12 n .
Multiplying by -n Next, we multiply the result, − 4 n 3 + 24 n 2 − 12 n , by − n . We distribute − n to each term: − n ( − 4 n 3 + 24 n 2 − 12 n ) = − n ( − 4 n 3 ) − n ( 24 n 2 ) − n ( − 12 n ) = 4 n 4 − 24 n 3 + 12 n 2 So, when this product is multiplied by − n , the result is 4 n 4 − 24 n 3 + 12 n 2 .
Final Answer Therefore, the product of ( n 2 − 6 n + 3 ) and − 4 n is − 4 n 3 + 24 n 2 − 12 n , and when this product is multiplied by − n , the result is 4 n 4 − 24 n 3 + 12 n 2 .
Examples
Understanding polynomial multiplication is crucial in various fields like engineering, physics, and computer science. For instance, when designing a bridge, engineers use polynomial equations to model the load distribution and stress on different parts of the structure. Multiplying these polynomials helps them predict the overall stability and safety of the bridge under various conditions. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces for 3D models.
