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

1. Identify the expressions: The expression is composed of three parts:
- [tex]\(7x^2\)[/tex]
- [tex]\(2x^3 + 5\)[/tex]
- [tex]\(x^2 - 4x - 9\)[/tex]

2. Multiply the first two expressions:
- Multiply [tex]\(7x^2\)[/tex] by each term in [tex]\(2x^3 + 5\)[/tex]:
- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]

So, the result of multiplying the first two expressions is:
[tex]\[14x^5 + 35x^2\][/tex]

3. Multiply the result with the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
- Distribute each term from the expression [tex]\(14x^5 + 35x^2\)[/tex] with each term in the polynomial [tex]\(x^2 - 4x - 9\)[/tex]:

- Multiply through by [tex]\(14x^5\)[/tex]:
- [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]

- Multiply through by [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]

4. Combine like terms:
- Group the terms with the same power of [tex]\(x\)[/tex]:
[tex]\[
\begin{align*}
& 14x^7 \\
& - 56x^6 \\
& - 126x^5 \\
& + 35x^4 \\
& - 140x^3 \\
& - 315x^2 \\
\end{align*}
\][/tex]

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

This is the result of multiplying and expanding the given expression.

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