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Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will expand it step-by-step.
1. Distribute the first term:
Start by distributing [tex]\(7x^2\)[/tex] into the polynomial [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
[tex]\[
= 14x^5 + 35x^2
\][/tex]
2. Multiply by the second polynomial:
Now, take the result [tex]\((14x^5 + 35x^2)\)[/tex] and multiply it by the polynomial [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]
Then, 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 like terms:
Put together all the terms we've calculated:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is our expanded expression, or product:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
This result matches one of the given options, confirming it as correct.
1. Distribute the first term:
Start by distributing [tex]\(7x^2\)[/tex] into the polynomial [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
[tex]\[
= 14x^5 + 35x^2
\][/tex]
2. Multiply by the second polynomial:
Now, take the result [tex]\((14x^5 + 35x^2)\)[/tex] and multiply it by the polynomial [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]
Then, 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 like terms:
Put together all the terms we've calculated:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is our expanded expression, or product:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
This result matches one of the given options, confirming it as correct.
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