Answer :
- Distribute the monomial $-2x^3$ to each term in the trinomial $(9x^3 - 7x^2 + 4x)$.
- Multiply the coefficients and add the exponents for each term: $-2x^3(9x^3) = -18x^6$, $-2x^3(-7x^2) = 14x^5$, and $-2x^3(4x) = -8x^4$.
- Combine the resulting terms to get the simplified expression: $-18x^6 + 14x^5 - 8x^4$.
- The final answer is $\boxed{-18x^6 + 14x^5 - 8x^4}$.
### Explanation
1. Understanding the Problem
We need to multiply the monomial $-2x^3$ by the trinomial $(9x^3 - 7x^2 + 4x)$. This involves distributing the monomial to each term inside the parentheses.
2. Distributing the Monomial
We distribute $-2x^3$ to each term of the trinomial:
$$-2x^3(9x^3 - 7x^2 + 4x) = -2x^3(9x^3) - 2x^3(-7x^2) - 2x^3(4x)$$
Now, we multiply each term.
3. Performing the Multiplication
Multiplying the first term:
$$-2x^3(9x^3) = -2 \times 9 \times x^3 \times x^3 = -18x^{3+3} = -18x^6$$
Multiplying the second term:
$$-2x^3(-7x^2) = -2 \times -7 \times x^3 \times x^2 = 14x^{3+2} = 14x^5$$
Multiplying the third term:
$$-2x^3(4x) = -2 \times 4 \times x^3 \times x = -8x^{3+1} = -8x^4$$
So, we have: $$-18x^6 + 14x^5 - 8x^4$$
4. Simplifying the Expression
Combining the results, we get:
$$-2x^3(9x^3 - 7x^2 + 4x) = -18x^6 + 14x^5 - 8x^4$$
The expression is now simplified.
5. Final Answer
The final simplified expression is $-18x^6 + 14x^5 - 8x^4$.
### Examples
Understanding polynomial multiplication is crucial in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For instance, when calculating the trajectory of a projectile, polynomial multiplication helps determine the projectile's position at different points in time. Similarly, in electrical engineering, polynomial multiplication is used to analyze circuits and design filters. Mastering this concept provides a foundation for solving real-world problems involving rates of change, optimization, and system analysis.
- Multiply the coefficients and add the exponents for each term: $-2x^3(9x^3) = -18x^6$, $-2x^3(-7x^2) = 14x^5$, and $-2x^3(4x) = -8x^4$.
- Combine the resulting terms to get the simplified expression: $-18x^6 + 14x^5 - 8x^4$.
- The final answer is $\boxed{-18x^6 + 14x^5 - 8x^4}$.
### Explanation
1. Understanding the Problem
We need to multiply the monomial $-2x^3$ by the trinomial $(9x^3 - 7x^2 + 4x)$. This involves distributing the monomial to each term inside the parentheses.
2. Distributing the Monomial
We distribute $-2x^3$ to each term of the trinomial:
$$-2x^3(9x^3 - 7x^2 + 4x) = -2x^3(9x^3) - 2x^3(-7x^2) - 2x^3(4x)$$
Now, we multiply each term.
3. Performing the Multiplication
Multiplying the first term:
$$-2x^3(9x^3) = -2 \times 9 \times x^3 \times x^3 = -18x^{3+3} = -18x^6$$
Multiplying the second term:
$$-2x^3(-7x^2) = -2 \times -7 \times x^3 \times x^2 = 14x^{3+2} = 14x^5$$
Multiplying the third term:
$$-2x^3(4x) = -2 \times 4 \times x^3 \times x = -8x^{3+1} = -8x^4$$
So, we have: $$-18x^6 + 14x^5 - 8x^4$$
4. Simplifying the Expression
Combining the results, we get:
$$-2x^3(9x^3 - 7x^2 + 4x) = -18x^6 + 14x^5 - 8x^4$$
The expression is now simplified.
5. Final Answer
The final simplified expression is $-18x^6 + 14x^5 - 8x^4$.
### Examples
Understanding polynomial multiplication is crucial in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For instance, when calculating the trajectory of a projectile, polynomial multiplication helps determine the projectile's position at different points in time. Similarly, in electrical engineering, polynomial multiplication is used to analyze circuits and design filters. Mastering this concept provides a foundation for solving real-world problems involving rates of change, optimization, and system analysis.
