Answer :
To solve the problem of finding the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will follow these steps:
1. Identify the expressions: We are multiplying three expressions:
- [tex]\(7x^2\)[/tex]
- [tex]\(2x^3 + 5\)[/tex]
- [tex]\(x^2 - 4x - 9\)[/tex]
2. Multiply the first two expressions:
Start by multiplying the [tex]\(7x^2\)[/tex] with each term in the second expression [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
This results in the expression:
[tex]\[
14x^5 + 35x^2
\][/tex]
3. Multiply the result with the third expression:
Now, take the polynomial obtained in the previous step and multiply it by the third expression [tex]\((x^2 - 4x - 9)\)[/tex].
- Multiply [tex]\(14x^5\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/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]
- Multiply [tex]\(35x^2\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/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]
4. Combine all terms:
Collect all the terms obtained in the previous step and combine like terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This gives us the final result:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This polynomial is the product of the three given expressions.
1. Identify the expressions: We are multiplying three expressions:
- [tex]\(7x^2\)[/tex]
- [tex]\(2x^3 + 5\)[/tex]
- [tex]\(x^2 - 4x - 9\)[/tex]
2. Multiply the first two expressions:
Start by multiplying the [tex]\(7x^2\)[/tex] with each term in the second expression [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
This results in the expression:
[tex]\[
14x^5 + 35x^2
\][/tex]
3. Multiply the result with the third expression:
Now, take the polynomial obtained in the previous step and multiply it by the third expression [tex]\((x^2 - 4x - 9)\)[/tex].
- Multiply [tex]\(14x^5\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/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]
- Multiply [tex]\(35x^2\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/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]
4. Combine all terms:
Collect all the terms obtained in the previous step and combine like terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This gives us the final result:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This polynomial is the product of the three given expressions.
