Answer :
- First, multiply the first two terms: $(7x^2)(2x^3+5) = 14x^5 + 35x^2$.
- Then, multiply the result by the third term: $(14x^5 + 35x^2)(x^2-4x-9) = 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- The product of the given polynomials is $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- Therefore, 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 given the expression $(7x^2)(2x^3+5)(x^2-4x-9)$ and asked to find the product.
2. Multiplying the First Two Terms
First, we multiply $(7x^2)$ and $(2x^3+5)$.
$$(7x^2)(2x^3+5) = 7x^2 * 2x^3 + 7x^2 * 5 = 14x^5 + 35x^2$$
3. Multiplying the Result by the Third Term
Next, we multiply $(14x^5 + 35x^2)$ and $(x^2-4x-9)$.
$$(14x^5 + 35x^2)(x^2-4x-9) = 14x^5(x^2-4x-9) + 35x^2(x^2-4x-9)$$
Now we distribute:
$$14x^5(x^2-4x-9) = 14x^7 - 56x^6 - 126x^5$$
$$35x^2(x^2-4x-9) = 35x^4 - 140x^3 - 315x^2$$
Combining these gives:
$$14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$$
4. Final Result
Therefore, the 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, when designing a bridge, engineers use polynomials to model the load distribution and ensure the structure's stability. Similarly, in computer graphics, polynomials are used to create curves and surfaces for realistic rendering.
- Then, multiply the result by the third term: $(14x^5 + 35x^2)(x^2-4x-9) = 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- The product of the given polynomials is $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- Therefore, 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 given the expression $(7x^2)(2x^3+5)(x^2-4x-9)$ and asked to find the product.
2. Multiplying the First Two Terms
First, we multiply $(7x^2)$ and $(2x^3+5)$.
$$(7x^2)(2x^3+5) = 7x^2 * 2x^3 + 7x^2 * 5 = 14x^5 + 35x^2$$
3. Multiplying the Result by the Third Term
Next, we multiply $(14x^5 + 35x^2)$ and $(x^2-4x-9)$.
$$(14x^5 + 35x^2)(x^2-4x-9) = 14x^5(x^2-4x-9) + 35x^2(x^2-4x-9)$$
Now we distribute:
$$14x^5(x^2-4x-9) = 14x^7 - 56x^6 - 126x^5$$
$$35x^2(x^2-4x-9) = 35x^4 - 140x^3 - 315x^2$$
Combining these gives:
$$14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$$
4. Final Result
Therefore, the 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, when designing a bridge, engineers use polynomials to model the load distribution and ensure the structure's stability. Similarly, in computer graphics, polynomials are used to create curves and surfaces for realistic rendering.
