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Answer :
- Multiply the first two polynomials: $(7x^2)(2x^3+5) = 14x^5 + 35x^2$.
- Multiply the result by the third polynomial: $(14x^5 + 35x^2)(x^2-4x-9)$.
- Apply the distributive property and expand: $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- The final product is $\boxed{14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2}$.
### Explanation
1. Problem Setup
We are asked to find the product of the polynomials $(7x^2)(2x^3+5)(x^2-4x-9)$. To do this, we will multiply the polynomials together step by step.
2. Multiplying the First Two Polynomials
First, let's multiply the first two polynomials: $(7x^2)(2x^3+5)$. Using the distributive property, we get:
$$(7x^2)(2x^3) + (7x^2)(5) = 14x^5 + 35x^2$$
3. Multiplying by the Third Polynomial
Now, we multiply the result by the third polynomial: $(14x^5 + 35x^2)(x^2-4x-9)$. Again, we use the distributive property:
$$14x^5(x^2-4x-9) + 35x^2(x^2-4x-9)$$
Expanding this, we have:
$$14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$$
4. Final Result
So, the final product is $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
### 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.
- Multiply the result by the third polynomial: $(14x^5 + 35x^2)(x^2-4x-9)$.
- Apply the distributive property and expand: $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- The final product is $\boxed{14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2}$.
### Explanation
1. Problem Setup
We are asked to find the product of the polynomials $(7x^2)(2x^3+5)(x^2-4x-9)$. To do this, we will multiply the polynomials together step by step.
2. Multiplying the First Two Polynomials
First, let's multiply the first two polynomials: $(7x^2)(2x^3+5)$. Using the distributive property, we get:
$$(7x^2)(2x^3) + (7x^2)(5) = 14x^5 + 35x^2$$
3. Multiplying by the Third Polynomial
Now, we multiply the result by the third polynomial: $(14x^5 + 35x^2)(x^2-4x-9)$. Again, we use the distributive property:
$$14x^5(x^2-4x-9) + 35x^2(x^2-4x-9)$$
Expanding this, we have:
$$14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$$
4. Final Result
So, the final product is $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
### 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.
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