High School

What is the product of the following expression?

[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 :

Sure! Let's go through the steps to find the product of the expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex].

### Step 1: Distribute the First Two Expressions

First, multiply [tex]\(7x^2\)[/tex] with both terms inside the parentheses of [tex]\(2x^3 + 5\)[/tex]:

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

Calculate each term:

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

So, the expression becomes:

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

### Step 2: Multiply the Result with the Third Expression

Now multiply the resulting expression [tex]\((14x^5 + 35x^2)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]:

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

Distribute each term from the first polynomial over the second:

- [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]
- [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]

Combine all these terms:

### Final Answer

The expanded product of the three initial expressions is:

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

That's the polynomial you were looking for!

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