Answer :
- First, expand $(2x^3+5)(x^2-4x-9)$ to get $2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45$.
- Then, multiply the result by $7x^2$ to obtain $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- Compare the expanded form with the given options.
- The correct product 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 four possible answers. Our goal is to expand the expression and identify the correct product from the given options.
2. Expanding the Second and Third Terms
First, we expand the second and third terms: $(2x^3+5)(x^2-4x-9)$. We multiply each term in the first parenthesis by each term in the second parenthesis:
$(2x^3)(x^2) + (2x^3)(-4x) + (2x^3)(-9) + (5)(x^2) + (5)(-4x) + (5)(-9)$
$= 2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45$
3. Multiplying by the First Term
Next, we multiply the result by the first term, $7x^2$:
$(7x^2)(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45)$
$= (7x^2)(2x^5) + (7x^2)(-8x^4) + (7x^2)(-18x^3) + (7x^2)(5x^2) + (7x^2)(-20x) + (7x^2)(-45)$
$= 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$
4. Identifying the Correct Option
Finally, we compare our result, $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$, with the four options provided. We see that it matches the third option.
Therefore, the correct product is $14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2$.
### 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 designing circuits or analyzing the trajectory of a projectile, engineers and physicists use polynomial multiplication to predict system behavior and optimize performance. This skill also helps in creating accurate mathematical models for simulations and predictions.
- Then, multiply the result by $7x^2$ to obtain $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$.
- Compare the expanded form with the given options.
- The correct product 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 four possible answers. Our goal is to expand the expression and identify the correct product from the given options.
2. Expanding the Second and Third Terms
First, we expand the second and third terms: $(2x^3+5)(x^2-4x-9)$. We multiply each term in the first parenthesis by each term in the second parenthesis:
$(2x^3)(x^2) + (2x^3)(-4x) + (2x^3)(-9) + (5)(x^2) + (5)(-4x) + (5)(-9)$
$= 2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45$
3. Multiplying by the First Term
Next, we multiply the result by the first term, $7x^2$:
$(7x^2)(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45)$
$= (7x^2)(2x^5) + (7x^2)(-8x^4) + (7x^2)(-18x^3) + (7x^2)(5x^2) + (7x^2)(-20x) + (7x^2)(-45)$
$= 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$
4. Identifying the Correct Option
Finally, we compare our result, $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$, with the four options provided. We see that it matches the third option.
Therefore, the correct product is $14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2$.
### 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 designing circuits or analyzing the trajectory of a projectile, engineers and physicists use polynomial multiplication to predict system behavior and optimize performance. This skill also helps in creating accurate mathematical models for simulations and predictions.
