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 :

To determine the product of the expressions [tex]\((7 x^2)\)[/tex], [tex]\((2 x^3 + 5)\)[/tex], and [tex]\((x^2 - 4 x - 9)\)[/tex], we can start by multiplying these expressions step-by-step.

1. Multiply [tex]\(7 x^2\)[/tex] by [tex]\((2 x^3 + 5)\)[/tex]:

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

Apply the distributive property:

[tex]\[
7 x^2 \cdot 2 x^3 + 7 x^2 \cdot 5
\][/tex]

Calculate each term:

[tex]\[
14 x^5 + 35 x^2
\][/tex]

So, the intermediate product is:

[tex]\[
14 x^5 + 35 x^2
\][/tex]

2. Now multiply this intermediate result by [tex]\((x^2 - 4 x - 9)\)[/tex]:

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

We apply the distributive property (each term in the first polynomial must be multiplied by each term in the second polynomial):

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

Now, let's compute each term:

[tex]\[
14 x^5 \cdot x^2 = 14 x^7
\][/tex]
[tex]\[
14 x^5 \cdot (-4 x) = -56 x^6
\][/tex]
[tex]\[
14 x^5 \cdot (-9) = -126 x^5
\][/tex]
[tex]\[
35 x^2 \cdot x^2 = 35 x^4
\][/tex]
[tex]\[
35 x^2 \cdot (-4 x) = -140 x^3
\][/tex]
[tex]\[
35 x^2 \cdot (-9) = -315 x^2
\][/tex]

Putting it all together:

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

Therefore, the product of the expressions is:

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

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