Answer :
- Distribute $7x^3$ to both terms inside the parenthesis: $7x^3 \times 6x^2 - 7x^3 \times 4x$.
- Simplify the first term: $7x^3 \times 6x^2 = 42x^5$.
- Simplify the second term: $7x^3 \times 4x = 28x^4$.
- The final expression is: $\boxed{42x^5 - 28x^4}$.
### Explanation
1. Understanding the problem
We are asked to find the product of the expression $7 x^3(6 x^2 - 4x)$. This involves distributing the term $7x^3$ to each term inside the parentheses.
2. Distributing the term
We distribute $7x^3$ to both terms inside the parenthesis: $7x^3 \times 6x^2 - 7x^3 \times 4x$.
3. Simplifying the expression
Now, we simplify each term by multiplying the coefficients and adding the exponents of $x$. For the first term, $7x^3 \times 6x^2 = (7 \times 6)x^{3+2} = 42x^5$. For the second term, $7x^3 \times 4x = (7 \times 4)x^{3+1} = 28x^4$. Therefore, the expression simplifies to $42x^5 - 28x^4$.
4. Choosing the correct option
Comparing our simplified expression $42x^5 - 28x^4$ with the given options, we see that it matches option C.
5. Final Answer
Therefore, the product of $7 x^3(6 x^2-4 x)$ is $42x^5 - 28x^4$.
### Examples
Understanding how to multiply polynomials is essential in various fields, such as engineering, physics, and computer graphics. For instance, when calculating the volume of a 3D object with polynomial dimensions or modeling physical phenomena with polynomial equations, polynomial multiplication is a fundamental operation. In computer graphics, it can be used to scale and transform objects.
- Simplify the first term: $7x^3 \times 6x^2 = 42x^5$.
- Simplify the second term: $7x^3 \times 4x = 28x^4$.
- The final expression is: $\boxed{42x^5 - 28x^4}$.
### Explanation
1. Understanding the problem
We are asked to find the product of the expression $7 x^3(6 x^2 - 4x)$. This involves distributing the term $7x^3$ to each term inside the parentheses.
2. Distributing the term
We distribute $7x^3$ to both terms inside the parenthesis: $7x^3 \times 6x^2 - 7x^3 \times 4x$.
3. Simplifying the expression
Now, we simplify each term by multiplying the coefficients and adding the exponents of $x$. For the first term, $7x^3 \times 6x^2 = (7 \times 6)x^{3+2} = 42x^5$. For the second term, $7x^3 \times 4x = (7 \times 4)x^{3+1} = 28x^4$. Therefore, the expression simplifies to $42x^5 - 28x^4$.
4. Choosing the correct option
Comparing our simplified expression $42x^5 - 28x^4$ with the given options, we see that it matches option C.
5. Final Answer
Therefore, the product of $7 x^3(6 x^2-4 x)$ is $42x^5 - 28x^4$.
### Examples
Understanding how to multiply polynomials is essential in various fields, such as engineering, physics, and computer graphics. For instance, when calculating the volume of a 3D object with polynomial dimensions or modeling physical phenomena with polynomial equations, polynomial multiplication is a fundamental operation. In computer graphics, it can be used to scale and transform objects.
