College

What is the product?

[tex]
\[
(7x^2)(2x^3 + 5)(x^2 - 4x - 9)
\]
[/tex]

A. [tex]\(14x^5 - x^4 - 46x^3 - 58x^2 - 20x - 45\)[/tex]

B. [tex]\(14x^6 - 56x^5 - 91x^4 - 140x^3 - 315x^2\)[/tex]

C. [tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex]

D. [tex]\(14x^{12} - 182x^6 + 35x^4 - 455x^2\)[/tex]

Answer :

Sure! To find the product of the given expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we'll follow these steps:

1. Distribute [tex]\(7x^2\)[/tex] into the second polynomial [tex]\((2x^3 + 5)\)[/tex]:

[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \times 2x^3 + 7x^2 \times 5
\][/tex]

Calculate:
- [tex]\(7x^2 \times 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]

So, after this step, we have the expression:
[tex]\[
14x^5 + 35x^2
\][/tex]

2. Multiply the result from the previous step by the third polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:

[tex]\((14x^5 + 35x^2) \times (x^2 - 4x - 9)\)[/tex]

Distribute each term of the first polynomial over each term of the second polynomial:

- [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]

- [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 the terms:

[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

So, the final result of the product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This is the expanded form of the product of the given expression.

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