Answer :
- First, multiply the first two polynomials: $(7x^2)(2x^3 + 5) = 14x^5 + 35x^2$.
- Then, multiply the result by the third polynomial: $(14x^5 + 35x^2)(x^2 - 4x - 9)$.
- Expand the expression: $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- The final answer is: $\boxed{14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2}$
### Explanation
1. Understanding the Problem
We are asked to find the product of three polynomials: $(7x^2)$, $(2x^3 + 5)$, and $(x^2 - 4x - 9)$. Our goal is to multiply these polynomials together and identify the correct result from the given options.
2. Multiplying the First Two Polynomials
First, let's multiply $(7x^2)$ and $(2x^3 + 5)$.
$$(7x^2)(2x^3 + 5) = 7x^2 * 2x^3 + 7x^2 * 5 = 14x^5 + 35x^2$$
So, we have $14x^5 + 35x^2$.
3. Multiplying by the Third Polynomial
Now, we multiply the result $(14x^5 + 35x^2)$ by the third polynomial $(x^2 - 4x - 9)$.
$$(14x^5 + 35x^2)(x^2 - 4x - 9) = 14x^5(x^2 - 4x - 9) + 35x^2(x^2 - 4x - 9)$$
Let's expand this expression:
$$14x^5(x^2 - 4x - 9) = 14x^7 - 56x^6 - 126x^5$$
$$35x^2(x^2 - 4x - 9) = 35x^4 - 140x^3 - 315x^2$$
Combining these, we get:
$$14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$$
4. Identifying the Correct Option
Comparing our result with the given options, we find that the correct answer is:
$$14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2$$
5. Final Answer
Therefore, the product of the given polynomials is:
$$\boxed{14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2}$$
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, the transfer function of a system can be represented as a ratio of two polynomials. Multiplying these polynomials helps in analyzing the system's behavior. Similarly, in computer graphics, polynomial multiplication is used in curve and surface modeling to create smooth and realistic shapes.
- Then, multiply the result by the third polynomial: $(14x^5 + 35x^2)(x^2 - 4x - 9)$.
- Expand the expression: $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- The final answer is: $\boxed{14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2}$
### Explanation
1. Understanding the Problem
We are asked to find the product of three polynomials: $(7x^2)$, $(2x^3 + 5)$, and $(x^2 - 4x - 9)$. Our goal is to multiply these polynomials together and identify the correct result from the given options.
2. Multiplying the First Two Polynomials
First, let's multiply $(7x^2)$ and $(2x^3 + 5)$.
$$(7x^2)(2x^3 + 5) = 7x^2 * 2x^3 + 7x^2 * 5 = 14x^5 + 35x^2$$
So, we have $14x^5 + 35x^2$.
3. Multiplying by the Third Polynomial
Now, we multiply the result $(14x^5 + 35x^2)$ by the third polynomial $(x^2 - 4x - 9)$.
$$(14x^5 + 35x^2)(x^2 - 4x - 9) = 14x^5(x^2 - 4x - 9) + 35x^2(x^2 - 4x - 9)$$
Let's expand this expression:
$$14x^5(x^2 - 4x - 9) = 14x^7 - 56x^6 - 126x^5$$
$$35x^2(x^2 - 4x - 9) = 35x^4 - 140x^3 - 315x^2$$
Combining these, we get:
$$14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$$
4. Identifying the Correct Option
Comparing our result with the given options, we find that the correct answer is:
$$14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2$$
5. Final Answer
Therefore, the product of the given polynomials is:
$$\boxed{14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2}$$
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, the transfer function of a system can be represented as a ratio of two polynomials. Multiplying these polynomials helps in analyzing the system's behavior. Similarly, in computer graphics, polynomial multiplication is used in curve and surface modeling to create smooth and realistic shapes.
