Answer :
To find the product of the expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], follow these steps:
1. Multiply the first two expressions:
Start by multiplying [tex]\(7x^2\)[/tex] with [tex]\(2x^3 + 5\)[/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]
- Calculate each part:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
- Combine these results:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third expression:
Now multiply the result [tex]\(14x^5 + 35x^2\)[/tex] by [tex]\(x^2 - 4x - 9\)[/tex].
Distribute each term in the first result to the entire third expression:
- [tex]\(14x^5 \cdot (x^2 - 4x - 9)\)[/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]
- [tex]\(35x^2 \cdot (x^2 - 4x - 9)\)[/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]
3. Combine all the terms:
- Put all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This expanded polynomial, [tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex], is the product of the three expressions given.
1. Multiply the first two expressions:
Start by multiplying [tex]\(7x^2\)[/tex] with [tex]\(2x^3 + 5\)[/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]
- Calculate each part:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
- Combine these results:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third expression:
Now multiply the result [tex]\(14x^5 + 35x^2\)[/tex] by [tex]\(x^2 - 4x - 9\)[/tex].
Distribute each term in the first result to the entire third expression:
- [tex]\(14x^5 \cdot (x^2 - 4x - 9)\)[/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]
- [tex]\(35x^2 \cdot (x^2 - 4x - 9)\)[/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]
3. Combine all the terms:
- Put all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This expanded polynomial, [tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex], is the product of the three expressions given.
