Answer :
Sure! Let's go through the steps to find the product of the following expression:
[tex]\[
\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right).
\][/tex]
### Step 1: Distribute the first binomial
First, distribute [tex]\(7x^2\)[/tex] to each term in the second binomial:
[tex]\[
(7x^2) \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2.
\][/tex]
So the expression now looks like this:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9).
\][/tex]
### Step 2: Distribute across the remaining binomial
Next, we need to distribute [tex]\( (14x^5 + 35x^2) \)[/tex] to each term in the third binomial [tex]\( (x^2 - 4x - 9) \)[/tex].
[tex]\[
\begin{aligned}
& (14x^5 + 35x^2) \cdot (x^2 - 4x - 9) = \\
& 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9) + 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9).
\end{aligned}
\][/tex]
Breaking this down step-by-step:
1. [tex]\(14x^5 \cdot x^2 = 14x^{7}\)[/tex]
2. [tex]\(14x^5 \cdot (-4x) = -56x^{6}\)[/tex]
3. [tex]\(14x^5 \cdot (-9) = -126x^{5}\)[/tex]
4. [tex]\(35x^2 \cdot x^2 = 35x^{4}\)[/tex]
5. [tex]\(35x^2 \cdot (-4x) = -140x^{3}\)[/tex]
6. [tex]\(35x^2 \cdot (-9) = -315x^{2}\)[/tex]
### Step 3: Combine all the terms
Now let's combine all these products:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the product of [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex] is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
And that's the final simplified expression.
[tex]\[
\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right).
\][/tex]
### Step 1: Distribute the first binomial
First, distribute [tex]\(7x^2\)[/tex] to each term in the second binomial:
[tex]\[
(7x^2) \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2.
\][/tex]
So the expression now looks like this:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9).
\][/tex]
### Step 2: Distribute across the remaining binomial
Next, we need to distribute [tex]\( (14x^5 + 35x^2) \)[/tex] to each term in the third binomial [tex]\( (x^2 - 4x - 9) \)[/tex].
[tex]\[
\begin{aligned}
& (14x^5 + 35x^2) \cdot (x^2 - 4x - 9) = \\
& 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9) + 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9).
\end{aligned}
\][/tex]
Breaking this down step-by-step:
1. [tex]\(14x^5 \cdot x^2 = 14x^{7}\)[/tex]
2. [tex]\(14x^5 \cdot (-4x) = -56x^{6}\)[/tex]
3. [tex]\(14x^5 \cdot (-9) = -126x^{5}\)[/tex]
4. [tex]\(35x^2 \cdot x^2 = 35x^{4}\)[/tex]
5. [tex]\(35x^2 \cdot (-4x) = -140x^{3}\)[/tex]
6. [tex]\(35x^2 \cdot (-9) = -315x^{2}\)[/tex]
### Step 3: Combine all the terms
Now let's combine all these products:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the product of [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex] is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
And that's the final simplified expression.
