College

What is the product of the expression?

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

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

1. Multiply the first two polynomials:
[tex]\[
7x^2 \cdot (2x^3 + 5)
\][/tex]
Distribute [tex]\(7x^2\)[/tex] to each term inside the parentheses:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
So, the product of the first two polynomials is:
[tex]\[
14x^5 + 35x^2
\][/tex]

2. Multiply the result with the third polynomial:
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
We distribute each term of the first polynomial across each term of the second polynomial:

- [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]

- [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:
Collect all the terms obtained from the multiplications:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This is the simplified expression for the product of the given polynomials.

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