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Answer :
To find the product of the given polynomial expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], follow these steps:
1. Distribute the first term: [tex]\(7x^2\)[/tex]
Multiply [tex]\(7x^2\)[/tex] by each term in the second polynomial:
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \times 2x^3 + 7x^2 \times 5 = 14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third polynomial: [tex]\(x^2 - 4x - 9\)[/tex]
We now need to distribute [tex]\((14x^5 + 35x^2)\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex].
First, multiply each term of [tex]\((14x^5)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \times (-9) = -126x^5
\][/tex]
Then, multiply each term of [tex]\((35x^2)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \times (-9) = -315x^2
\][/tex]
3. Combine all terms from the multiplication
Combine all the terms to form the final expanded expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the product of the polynomials is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This matches the third option given in the list of choices.
1. Distribute the first term: [tex]\(7x^2\)[/tex]
Multiply [tex]\(7x^2\)[/tex] by each term in the second polynomial:
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \times 2x^3 + 7x^2 \times 5 = 14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third polynomial: [tex]\(x^2 - 4x - 9\)[/tex]
We now need to distribute [tex]\((14x^5 + 35x^2)\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex].
First, multiply each term of [tex]\((14x^5)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \times (-9) = -126x^5
\][/tex]
Then, multiply each term of [tex]\((35x^2)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \times (-9) = -315x^2
\][/tex]
3. Combine all terms from the multiplication
Combine all the terms to form the final expanded expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the product of the polynomials is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This matches the third option given in the list of choices.
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