Answer :
Sure! Let's go through finding the product step by step.
We need to find the product of the following three expressions:
[tex]\[
(7x^2)(2x^3 + 5)(x^2 - 4x - 9)
\][/tex]
To solve this, we can follow these steps:
1. Start by multiplying the first two expressions.
2. Then multiply the result by the third expression.
### Step 1: Multiply the first two expressions
First two expressions:
[tex]\[
7x^2 \quad \text{and} \quad 2x^3 + 5
\][/tex]
Distribute [tex]\( 7x^2 \)[/tex] to each term in the second expression:
[tex]\[
7x^2 \cdot (2x^3 + 5) = (7x^2 \cdot 2x^3) + (7x^2 \cdot 5)
\][/tex]
Multiply the terms:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5 \quad \text{and} \quad 7x^2 \cdot 5 = 35x^2
\][/tex]
So the result of the first multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result by the third expression
Now we have:
[tex]\[
(14x^5 + 35x^2) \quad \text{and} \quad (x^2 - 4x - 9)
\][/tex]
We'll use the distributive property again to multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9) = 14x^5 \cdot (x^2 - 4x - 9) + 35x^2 \cdot (x^2 - 4x - 9)
\][/tex]
Now, let's distribute each term separately:
#### Distributing [tex]\( 14x^5 \)[/tex]:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-9) = -126x^5
\][/tex]
#### Distributing [tex]\( 35x^2 \)[/tex]:
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-9) = -315x^2
\][/tex]
### Step 3: Combine all the terms
Now, we combine all the resulting terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the final product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the correct answer is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
We need to find the product of the following three expressions:
[tex]\[
(7x^2)(2x^3 + 5)(x^2 - 4x - 9)
\][/tex]
To solve this, we can follow these steps:
1. Start by multiplying the first two expressions.
2. Then multiply the result by the third expression.
### Step 1: Multiply the first two expressions
First two expressions:
[tex]\[
7x^2 \quad \text{and} \quad 2x^3 + 5
\][/tex]
Distribute [tex]\( 7x^2 \)[/tex] to each term in the second expression:
[tex]\[
7x^2 \cdot (2x^3 + 5) = (7x^2 \cdot 2x^3) + (7x^2 \cdot 5)
\][/tex]
Multiply the terms:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5 \quad \text{and} \quad 7x^2 \cdot 5 = 35x^2
\][/tex]
So the result of the first multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result by the third expression
Now we have:
[tex]\[
(14x^5 + 35x^2) \quad \text{and} \quad (x^2 - 4x - 9)
\][/tex]
We'll use the distributive property again to multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9) = 14x^5 \cdot (x^2 - 4x - 9) + 35x^2 \cdot (x^2 - 4x - 9)
\][/tex]
Now, let's distribute each term separately:
#### Distributing [tex]\( 14x^5 \)[/tex]:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-9) = -126x^5
\][/tex]
#### Distributing [tex]\( 35x^2 \)[/tex]:
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-9) = -315x^2
\][/tex]
### Step 3: Combine all the terms
Now, we combine all the resulting terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the final product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the correct answer is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
