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Answer :
To find the product of the three expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we'll follow a step-by-step approach to multiply them together.
### Step 1: Expand the first two expressions
Start by multiplying the first two expressions:
[tex]\[
(7x^2) \cdot (2x^3 + 5)
\][/tex]
Distribute [tex]\(7x^2\)[/tex] across each term in the second expression:
[tex]\[
= 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculate each term:
[tex]\[
= 14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result with the third expression
Now, take [tex]\((14x^5 + 35x^2)\)[/tex] and multiply it with the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
#### First, distribute [tex]\(14x^5\)[/tex]:
[tex]\[
= 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\][/tex]
[tex]\[
= 14x^7 - 56x^6 - 126x^5
\][/tex]
#### Next, distribute [tex]\(35x^2\)[/tex]:
[tex]\[
= 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
[tex]\[
= 35x^4 - 140x^3 - 315x^2
\][/tex]
### Step 3: Combine all terms
Now, bring all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
These terms are already simplified. Therefore, this is the final expanded expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the answer is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
### Step 1: Expand the first two expressions
Start by multiplying the first two expressions:
[tex]\[
(7x^2) \cdot (2x^3 + 5)
\][/tex]
Distribute [tex]\(7x^2\)[/tex] across each term in the second expression:
[tex]\[
= 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculate each term:
[tex]\[
= 14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result with the third expression
Now, take [tex]\((14x^5 + 35x^2)\)[/tex] and multiply it with the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
#### First, distribute [tex]\(14x^5\)[/tex]:
[tex]\[
= 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\][/tex]
[tex]\[
= 14x^7 - 56x^6 - 126x^5
\][/tex]
#### Next, distribute [tex]\(35x^2\)[/tex]:
[tex]\[
= 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
[tex]\[
= 35x^4 - 140x^3 - 315x^2
\][/tex]
### Step 3: Combine all terms
Now, bring all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
These terms are already simplified. Therefore, this is the final expanded expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the answer is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
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