4. $(-7 c^2 d^2)(-4 c d+6 c d^3-2 d)$
5. $(-x y z)(-6 x y+2 x^2 y^2 z-4 x y^2)$
6. $\frac{50 a b^3 c^4+40 a b^5 c^7-30 a^2 b^4 c^3}{10 a b^2 c^3}$
7. $\frac{-m^5 n^4+m^4 n^5-m^3 n^3}{-m^3 n^3}$

Multiply polynomials by distributing each term of one polynomial to each term of the other polynomial and simplifying.
Divide polynomials by dividing each term of the polynomial by the monomial and simplifying.
The result of ( − 7 c 2 d 2 ) ( − 4 c d + 6 c d 3 − 2 d ) is 28 c 3 d 3 − 42 c 3 d 5 + 14 c 2 d 3 .
The result of ( − x yz ) ( − 6 x y + 2 x 2 y 2 z − 4 x y 2 ) is 6 x 2 y 2 z − 2 x 3 y 3 z 2 + 4 x 2 y 3 z .
The result of 10 a b 2 c 3 50 a b 3 c 4 + 40 a b 5 c 7 − 30 a 2 b 4 c 3 ​ is 5 b c + 4 b 3 c 4 − 3 a b 2 .
The result of − m 3 n 3 − m 5 n 4 + m 4 n 5 − m 3 n 3 ​ is m 2 n − m n 2 + 1 .
The final answers are: 28 c 3 d 3 − 42 c 3 d 5 + 14 c 2 d 3 , 6 x 2 y 2 z − 2 x 3 y 3 z 2 + 4 x 2 y 3 z , 5 b c + 4 b 3 c 4 − 3 a b 2 , m 2 n − m n 2 + 1 .
28 c 3 d 3 − 42 c 3 d 5 + 14 c 2 d 3 , 6 x 2 y 2 z − 2 x 3 y 3 z 2 + 4 x 2 y 3 z , 5 b c + 4 b 3 c 4 − 3 a b 2 , m 2 n − m n 2 + 1 ​

Explanation

Introduction We will solve the given polynomial multiplication and division problems step by step.

Solving Problem 4 (Multiplication) Problem 4: Multiply the polynomial ( − 7 c 2 d 2 ) by the polynomial ( − 4 c d + 6 c d 3 − 2 d ) . We distribute ( − 7 c 2 d 2 ) to each term in ( − 4 c d + 6 c d 3 − 2 d ) .
( − 7 c 2 d 2 ) ( − 4 c d ) + ( − 7 c 2 d 2 ) ( 6 c d 3 ) + ( − 7 c 2 d 2 ) ( − 2 d ) = 28 c 3 d 3 − 42 c 3 d 5 + 14 c 2 d 3

Solving Problem 5 (Multiplication) Problem 5: Multiply the polynomial ( − x yz ) by the polynomial ( − 6 x y + 2 x 2 y 2 z − 4 x y 2 ) . We distribute ( − x yz ) to each term in ( − 6 x y + 2 x 2 y 2 z − 4 x y 2 ) .
( − x yz ) ( − 6 x y ) + ( − x yz ) ( 2 x 2 y 2 z ) + ( − x yz ) ( − 4 x y 2 ) = 6 x 2 y 2 z − 2 x 3 y 3 z 2 + 4 x 2 y 3 z

Solving Problem 4 (Division) Problem 4: Divide the polynomial ( 50 a b 3 c 4 + 40 a b 5 c 7 − 30 a 2 b 4 c 3 ) by the monomial ( 10 a b 2 c 3 ) . We divide each term in ( 50 a b 3 c 4 + 40 a b 5 c 7 − 30 a 2 b 4 c 3 ) by ( 10 a b 2 c 3 ) .
10 a b 2 c 3 50 a b 3 c 4 ​ + 10 a b 2 c 3 40 a b 5 c 7 ​ − 10 a b 2 c 3 30 a 2 b 4 c 3 ​ = 5 b c + 4 b 3 c 4 − 3 a b 2

Solving Problem 5 (Division) Problem 5: Divide the polynomial ( − m 5 n 4 + m 4 n 5 − m 3 n 3 ) by the monomial ( − m 3 n 3 ) . We divide each term in ( − m 5 n 4 + m 4 n 5 − m 3 n 3 ) by ( − m 3 n 3 ) .
− m 3 n 3 − m 5 n 4 ​ + − m 3 n 3 m 4 n 5 ​ − − m 3 n 3 m 3 n 3 ​ = m 2 n − m n 2 + 1

Final Answer Final Answers: Problem 4 (Multiplication): 28 c 3 d 3 − 42 c 3 d 5 + 14 c 2 d 3 Problem 5 (Multiplication): 6 x 2 y 2 z − 2 x 3 y 3 z 2 + 4 x 2 y 3 z Problem 4 (Division): 5 b c + 4 b 3 c 4 − 3 a b 2 Problem 5 (Division): m 2 n − m n 2 + 1


Examples
Polynomial multiplication and division are fundamental operations in algebra and are used in various fields such as engineering, physics, and computer science. For example, in computer graphics, polynomial functions are used to model curves and surfaces, and performing operations on these polynomials allows for manipulating and rendering complex shapes. In physics, polynomial equations can describe the motion of objects, and polynomial division can be used to simplify these equations to analyze the system more easily. Understanding these operations is crucial for solving real-world problems in these fields.

Answered by GinnyAnswer | 2025-07-07