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Answer :
To find the product [tex]\((7x^2)(2x^3+5)(x^2-4x-9)\)[/tex], we'll take it step by step. Let's start simplifying by multiplying the first two polynomials and then multiplying the result by the third polynomial.
1. Multiply [tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex]:
Distribute [tex]\(7x^2\)[/tex] across each term in [tex]\(2x^3 + 5\)[/tex]:
[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. Now, multiply the result by [tex]\((x^2 - 4x - 9)\)[/tex]:
Distribute [tex]\(14x^5\)[/tex] across 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]
Now, distribute [tex]\(35x^2\)[/tex] across 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]
3. Combine all the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the final expanded product of the given expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This matches the option:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
1. Multiply [tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex]:
Distribute [tex]\(7x^2\)[/tex] across each term in [tex]\(2x^3 + 5\)[/tex]:
[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. Now, multiply the result by [tex]\((x^2 - 4x - 9)\)[/tex]:
Distribute [tex]\(14x^5\)[/tex] across 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]
Now, distribute [tex]\(35x^2\)[/tex] across 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]
3. Combine all the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the final expanded product of the given expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This matches the option:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
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