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! Let's find the product of the given expressions step-by-step:

The expressions given are:
[tex]\[
(7x^2), \quad (2x^3 + 5), \quad \text{and} \quad (x^2 - 4x - 9)
\][/tex]

We need to multiply these expressions together to find their product. Here's how we can proceed:

Step 1: Distribute and multiply the first two expressions

First, multiply [tex]\( 7x^2 \)[/tex] by each term inside the second expression [tex]\( (2x^3 + 5) \)[/tex]:

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

Simplifying each term, we get:

[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]

Hence, the product of the first two expressions is:

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

Step 2: Multiply the resulting expression by the third expression

Next, we need to multiply [tex]\( (14x^5 + 35x^2) \)[/tex] by the third expression [tex]\( (x^2 - 4x - 9) \)[/tex]:

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

Distribute each term from the first expression to each term of the second expression:

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

Now, let's handle these multiplications separately:

[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]

Therefore, the product of [tex]\( 14x^5 \)[/tex] with [tex]\( x^2 - 4x - 9 \)[/tex] is:

[tex]\[
14x^7 - 56x^6 - 126x^5
\][/tex]

Next, perform the similar operation for [tex]\( 35x^2 \)[/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]

Therefore, the product of [tex]\( 35x^2 \)[/tex] with [tex]\( x^2 - 4x - 9 \)[/tex] is:

[tex]\[
35x^4 - 140x^3 - 315x^2
\][/tex]

Step 3: Combine all the terms

Add up all the results from the individual terms:

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

This yields:

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

Therefore, the final product of the given expressions is:

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

From the options given, this matches the third option:

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