High School

What is the product?

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

B. [tex]14x^5-x^4-46x^3-58x^2-20x-45[/tex]

C. [tex]14x^6-56x^5-91x^4-140x^3-315x^2[/tex]

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

E. [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'll tackle it step-by-step by multiplying these expressions together.

1. Multiply the first two expressions: [tex]\((7x^2)(2x^3 + 5)\)[/tex]

- Distribute [tex]\(7x^2\)[/tex] across each term of the polynomial [tex]\(2x^3 + 5\)[/tex].

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

- Adding these, we have:

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

2. Multiply the result by the third expression: [tex]\((14x^5 + 35x^2)(x^2 - 4x - 9)\)[/tex]

- Distribute each term in the first polynomial across all the terms in the second polynomial:

a. Multiply [tex]\(14x^5\)[/tex] by each term in [tex]\(x^2 - 4x - 9\)[/tex]:

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

b. Multiply [tex]\(35x^2\)[/tex] by each term in [tex]\(x^2 - 4x - 9\)[/tex]:

[tex]\[
(35x^2)(x^2) = 35x^4
\][/tex]
[tex]\[
(35x^2)(-4x) = -140x^3
\][/tex]
[tex]\[
(35x^2)(-9) = -315x^2
\][/tex]

3. Combine all terms:

- Add up all the terms:

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

This is the expanded form of the product of the original expressions. Thus, the product is:

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

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