We appreciate your visit to What is the product of the expression tex left 7x 2 right left 2x 3 5 right left x 2 4x 9 right tex A. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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.
Thanks for taking the time to read What is the product of the expression tex left 7x 2 right left 2x 3 5 right left x 2 4x 9 right tex A. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
