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Answer :
To find the product of the given expression [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex], we follow these steps:
1. Distribute the first part [tex]\((7x^2)\)[/tex] to the second part [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
(7x^2) \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
This results in:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Now multiply this result by the third part [tex]\((x^2 - 4x - 9)\)[/tex]:
We distribute each term from [tex]\(14x^5 + 35x^2\)[/tex] across [tex]\(x^2 - 4x - 9\)[/tex].
- First, distribute [tex]\(14x^5\)[/tex]:
[tex]\[
14x^5 (x^2 - 4x - 9) = 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\][/tex]
[tex]\[
= 14x^7 - 56x^6 - 126x^5
\][/tex]
- Next, distribute [tex]\(35x^2\)[/tex]:
[tex]\[
35x^2 (x^2 - 4x - 9) = 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
[tex]\[
= 35x^4 - 140x^3 - 315x^2
\][/tex]
3. Combine all terms:
- Combine the terms from both distributions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded expression, which is the product of the given expressions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the final answer.
1. Distribute the first part [tex]\((7x^2)\)[/tex] to the second part [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
(7x^2) \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
This results in:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Now multiply this result by the third part [tex]\((x^2 - 4x - 9)\)[/tex]:
We distribute each term from [tex]\(14x^5 + 35x^2\)[/tex] across [tex]\(x^2 - 4x - 9\)[/tex].
- First, distribute [tex]\(14x^5\)[/tex]:
[tex]\[
14x^5 (x^2 - 4x - 9) = 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\][/tex]
[tex]\[
= 14x^7 - 56x^6 - 126x^5
\][/tex]
- Next, distribute [tex]\(35x^2\)[/tex]:
[tex]\[
35x^2 (x^2 - 4x - 9) = 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
[tex]\[
= 35x^4 - 140x^3 - 315x^2
\][/tex]
3. Combine all terms:
- Combine the terms from both distributions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded expression, which is the product of the given expressions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the final answer.
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