Answer :
To find the product [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 8)\)[/tex], we will multiply these polynomial expressions step by step.
1. Multiply the first two expressions: [tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] across the terms inside the parentheses:
[tex]\[
7x^2 \cdot (2x^3 + 5) = (7x^2 \cdot 2x^3) + (7x^2 \cdot 5)
\][/tex]
- Calculate each term:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
- Combine these results:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third expression: [tex]\((x^2 - 4x - 8)\)[/tex].
- Distribute each term from [tex]\(14x^5 + 35x^2\)[/tex] to each term in [tex]\(x^2 - 4x - 8\)[/tex].
First, distribute [tex]\(14x^5\)[/tex]:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-8) = -112x^5
\][/tex]
Next, distribute [tex]\(35x^2\)[/tex]:
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-8) = -280x^2
\][/tex]
3. Combine all the terms obtained from distribution:
Collect all terms from each multiplication step:
[tex]\[
14x^7 - 56x^6 - 112x^5 + 35x^4 - 140x^3 - 280x^2
\][/tex]
This is the expanded polynomial product of the given expressions. So the product is:
[tex]\[
14x^7 - 56x^6 - 112x^5 + 35x^4 - 140x^3 - 280x^2
\][/tex]
1. Multiply the first two expressions: [tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] across the terms inside the parentheses:
[tex]\[
7x^2 \cdot (2x^3 + 5) = (7x^2 \cdot 2x^3) + (7x^2 \cdot 5)
\][/tex]
- Calculate each term:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
- Combine these results:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third expression: [tex]\((x^2 - 4x - 8)\)[/tex].
- Distribute each term from [tex]\(14x^5 + 35x^2\)[/tex] to each term in [tex]\(x^2 - 4x - 8\)[/tex].
First, distribute [tex]\(14x^5\)[/tex]:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-8) = -112x^5
\][/tex]
Next, distribute [tex]\(35x^2\)[/tex]:
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-8) = -280x^2
\][/tex]
3. Combine all the terms obtained from distribution:
Collect all terms from each multiplication step:
[tex]\[
14x^7 - 56x^6 - 112x^5 + 35x^4 - 140x^3 - 280x^2
\][/tex]
This is the expanded polynomial product of the given expressions. So the product is:
[tex]\[
14x^7 - 56x^6 - 112x^5 + 35x^4 - 140x^3 - 280x^2
\][/tex]
