High School

What is the product?

[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 :

We start with the expression

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

Step 1: Multiply the first two factors.

Multiply [tex]\(7x^2\)[/tex] by each term in [tex]\(2x^3 + 5\)[/tex]:

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

So, the product of the first two factors is

[tex]$$
14x^5 + 35x^2.
$$[/tex]

Step 2: Multiply the resulting expression by the third factor.

Now multiply

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

by distributing each term in the first polynomial to every term in the second polynomial:

- Multiply [tex]\(14x^5\)[/tex] by each term:
[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 [tex]\(35x^2\)[/tex] by each term:
[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]

Step 3: Combine all the terms.

Putting all of these products together, we have

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

Thus, the fully expanded product is

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

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