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Answer :
To find the product of the polynomials [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will follow a step-by-step approach:
1. Distribute the first polynomial: Start by expanding [tex]\(7x^2\)[/tex] with the second polynomial [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third polynomial: Distribute every term from the expanded result above with each term in [tex]\((x^2 - 4x - 9)\)[/tex].
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
3. Expand each product:
For [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]
For [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]
4. Combine all terms:
Now, put together all the results from above expansions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This simplified expression looks exactly like the third choice in the given options:
[tex]\[ \boxed{3: \, 14x^7 - 56x^6 - 126x^5 + 85x^4 - 140x^3 - 315x^2} \][/tex]
Thus, the answer to the problem is the third option.
1. Distribute the first polynomial: Start by expanding [tex]\(7x^2\)[/tex] with the second polynomial [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third polynomial: Distribute every term from the expanded result above with each term in [tex]\((x^2 - 4x - 9)\)[/tex].
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
3. Expand each product:
For [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]
For [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]
4. Combine all terms:
Now, put together all the results from above expansions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This simplified expression looks exactly like the third choice in the given options:
[tex]\[ \boxed{3: \, 14x^7 - 56x^6 - 126x^5 + 85x^4 - 140x^3 - 315x^2} \][/tex]
Thus, the answer to the problem is the third option.
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