Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will multiply the terms step-by-step. Let's break it down:
### Step 1: Multiply the first two expressions
First, we'll multiply [tex]\(7x^2\)[/tex] with [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculating each term separately:
- [tex]\(7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, the product of these two expressions is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result with the third expression
Now, multiply the result from Step 1, [tex]\((14x^5 + 35x^2)\)[/tex], with [tex]\((x^2 - 4x - 9)\)[/tex].
[tex]\((14x^5 + 35x^2) \times (x^2 - 4x - 9)\)[/tex] can be expanded as follows:
1. Distribute [tex]\(14x^5\)[/tex]:
- [tex]\(14x^5 \cdot x^2 = 14x^{5+2} = 14x^7\)[/tex]
- [tex]\(14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6\)[/tex]
- [tex]\(14x^5 \cdot (-9) = -126x^5\)[/tex]
2. Distribute [tex]\(35x^2\)[/tex]:
- [tex]\(35x^2 \cdot x^2 = 35x^{2+2} = 35x^4\)[/tex]
- [tex]\(35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3\)[/tex]
- [tex]\(35x^2 \cdot (-9) = -315x^2\)[/tex]
### Step 3: Combine all terms
Add all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the product of the given expressions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This completes the multiplication, and this expression is the final result.
### Step 1: Multiply the first two expressions
First, we'll multiply [tex]\(7x^2\)[/tex] with [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculating each term separately:
- [tex]\(7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, the product of these two expressions is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result with the third expression
Now, multiply the result from Step 1, [tex]\((14x^5 + 35x^2)\)[/tex], with [tex]\((x^2 - 4x - 9)\)[/tex].
[tex]\((14x^5 + 35x^2) \times (x^2 - 4x - 9)\)[/tex] can be expanded as follows:
1. Distribute [tex]\(14x^5\)[/tex]:
- [tex]\(14x^5 \cdot x^2 = 14x^{5+2} = 14x^7\)[/tex]
- [tex]\(14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6\)[/tex]
- [tex]\(14x^5 \cdot (-9) = -126x^5\)[/tex]
2. Distribute [tex]\(35x^2\)[/tex]:
- [tex]\(35x^2 \cdot x^2 = 35x^{2+2} = 35x^4\)[/tex]
- [tex]\(35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3\)[/tex]
- [tex]\(35x^2 \cdot (-9) = -315x^2\)[/tex]
### Step 3: Combine all terms
Add all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the product of the given expressions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This completes the multiplication, and this expression is the final result.
